Trigonometric Functions
Then
Now
Why?
Throughout this text, you have graphed and analyzed functions.
In Chapter 13, you will:
WATER SPORTS Knowing trigonometric functions has practical applications in water sports. For instance, you can use right triangle trigonometry to find the distance a kayak has traveled down river. If you are familiar with angles and angle measures, then you have a better understanding of how impressive it is to be able to do a 540° rotation on a wakeboard.
Find values of trigonometric functions. Solve problems by using right triangle trigonometry. Solve triangles by using the Law of Sines and Law of Cosines. Graph trigonometric functions.
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Tennessee Curriculum Standards CLE 3103.4.5
Get Ready for the Chapter Diagnose Readiness
1

You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
QuickCheck
QuickReview
Find the value of x to the nearest tenth. (Lesson 07) 1.
x
Example 1 Find the missing measure of the right triangle.
4
18
11
2.
c2 = a2 + b2
x
12
a
8
3. x
5
2
22
2
18 = a + 5
2
324 = a 2 + 25
9
299 = a 4. Laura has a rectangular garden in her backyard that measures 12 feet by 15 feet. She wants to put a rock walkway on the diagonal. How long will the walkway be? Round to the nearest tenth of a foot. Find each missing measure. Write all radicals in simplest form. (Geometry)
2
Pythagorean Theorem Replace c with 18 and b with 5. Simplify. Subtract 25 from each side.
17.3 ≈ a
Take the positive square root of each side.
Example 2 Find the missing measures. Write all radicals in simplest form.
5. y
x
18 45°
45°
9
6.
x
60°
7.
x
2
y
30°
2
x + x = 18
8 y
2
2x 2 = 18 2
30°
24 60° x
8. A ladder leans against a wall at a 45° angle. If the ladder is 12 feet long, how far up the wall does the ladder reach?
2
x
2
2x = 324 x 2 = 162
Pythagorean Theorem Combine like terms. Simplify. Divide each side by 2.
x = √ 162
Take the positive square root of each side.
x = 9 √ 2
Simplify.
Online Option Take an online selfcheck Chapter Readiness Quiz at connectED.mcgrawhill.com. 805
Get Started on the Chapter You will learn several new concepts, skills, and vocabulary terms as you study Chapter 13. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills.
StudyOrganizer
NewVocabulary
Trigonometric Functions Make this Foldable to help you organize your Chapter 13 notes about trigonometric functions. Begin with four pieces of grid paper.
2.5 in.
2
3
4
Stack paper together and measure 2.5 inches from the bottom.
Fold on the diagonal.
Staple along the diagonal to form a book.
Label the edge as Trigonometric Functions.
Trignonometric Functions
1
English
Español
trigonometry
p. 808
trigonometría
sine
p. 808
seno
cosine
p. 808
coseno
tangent
p. 808
tangente
cosecant
p. 808
cosecante
secant
p. 808
secante
cotangent
p. 808
cotangente
angle of elevation
p. 812
ángulo de depresión
angle of depression
p. 812
ángulo de elevación
standard position
p. 817
posición estándar
radian
p. 819
radián
Law of Sines
p. 833
Ley de los senos
ambiguous case
p. 836
caso ambiguo
Law of Cosines
p. 841
Ley de los cosenos
unit circle
p. 848
círculo unitario
circular function
p. 848
funciones circulares
periodic function
p. 849
función periódica
cycle
p. 849
ciclo
period
p. 849
período
amplitude frequency
p. 855 p. 856
amplitud frecuencia
ReviewVocabulary 131
806  Chapter 13  Trigonometric Functions
acute angle prior course ángulo agudo an angle with a measure between 0˚ and 90˚ function p. P4 función a relation in which each element of the domain is paired with exactly one element in the range inverse function p. 417 función inversa two functions f and g are inverse functions if and only if both of their compositions are the identity function
Spreadsheet Lab
Investigating Special Right Triangles You can use a spreadsheet to investigate side measures of special right triangles.
Activity
45°45°90° Triangle
The legs of a 45°45°90° triangle, a and b, are equal in measure. What patterns do you observe in the ratios of the side measures of these triangles?
" 45° c
Step 1 Enter the indicated formulas in the spreadsheet. The formula uses the Pythagorean Theorem in the form c = √ a2 + b2 . =SQRT(A2^2+B2^2)
=B2/A2
45°
=B2/C2
=A2/C2
b
a
#
454590 triangles 1 2 3 4 5
$
A
B
C
D
E
F
Tennessee Curriculum Standards
a 1 2 3 4
b 1 2 3 4
c 1.414213562 2.828427125 4.242640687 5.656854249
b/a 1 1 1 1
b/c 0.707106781 0.707106781 0.707106781 0.707106781
a/c 0.707106781 0.707106781 0.707106781 0.707106781
✔ 3103.5.2 Organize and display data using appropriate methods (including spreadsheets and technology tools) to detect patterns and departures from patterns.
Sheet 1
Sheet 2
Sheet 3
Step 2 Examine the results. Because 45°45°90° triangles share the same angle measures, these triangles are all similar. The ratios of the sides of these triangles are all the same. The ratios of side b to side a are 1. The ratios of side b to side c and of side a to side c are approximately 0.71.
Model and Analyze #
Use the spreadsheet below for 30°60°90° triangles. c
USJBOHMFT
"
#
$
%
&
'
B
C
D
CB
CD
BD
4IFFU
4IFFU
30°
"
b
60°
a
$
4IFFU
1. Copy and complete the spreadsheet above. 2. Describe the relationship among the 30°60°90° triangles with the dimensions given. 3. What patterns do you observe in the ratios of the side measures of these triangles? connectED.mcgrawhill.com
807
Trigonometric Functions in Right Triangles Then
Now
Why?
You used the Pythagorean Theorem to find side lengths of right triangles.
1
Find values of trigonometric functions for acute angles.
2
Use trigonometric functions to find side lengths and angle measures of right triangles.
The altitude of a person parasailing depends on the length of the tow rope and the angle the rope makes with the horizontal x°. If you know these two values, you can use a ratio to find the altitude of the person parasailing.
NewVocabulary trigonometry trigonometric ratio trigonometric function sine cosine tangent cosecant secant cotangent reciprocal functions angle of elevation angle of depression
1
x°
Trigonometric Functions for Acute Angles Trigonometry is the study of
relationships among the angles and sides of a right triangle. A trigonometric ratio compares the side lengths of a right triangle. A trigonometric function has a rule given by a trigonometric ratio. "
The Greek letter theta θ is often used to represent the measure of an acute angle in a right triangle. The hypotenuse, the leg opposite θ, and the leg adjacent to θ are used to define the six trigonometric functions.
opposite leg
$
hypotenuse θ adjacent leg
#
KeyConcept Trigonometric Functions in Right Triangles Words
If θ is the measure of an acute angle of a right triangle, then the following trigonometric functions involving the opposite side opp, the adjacent side adj, and the hypotenuse hyp are true.
Symbols
sin (sine) θ = _
Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
altitude
Examples
opp hyp adj cos (cosine) θ = _ hyp opp tan (tangent) θ = _ adj
_ 5 5 csc θ = _ sin θ = 4
4
hyp
csc (cosecant) θ = _ opp hyp adj adj cot (cotangent) θ = _ opp
sec (secant) θ = _
_ 5 _ sec θ = 5 cos θ = 3 3
5
4
θ 3
4 tan θ = _ 3
3 cot θ = _ 4
Example 1 Evaluate Trigonometric Functions Find the values of the six trigonometric functions for angle θ. leg opposite θ: BC = 8
leg adjacent θ: AC = 15 hypotenuse: AB = 17
opp 8 sin θ = _ = _ 17 hyp hyp 17 _ csc θ = _ opp = 8
adj 15 cos θ = _ = _ 17 hyp hyp 17 sec θ = _ = _ 15 adj
opp 8 tan θ = _ = _ 15 adj adj 15 _ cot θ = _ opp = 8 "
GuidedPractice 1. Find the values of the six trigonometric functions for angle B.
808  Lesson 131
# 17
8
θ 15
$
StudyTip Memorize Trigonometric Ratios SOHCAHTOA is a mnemonic device for remembering the first letter of each word in the ratios for sine, cosine, and tangent. opp hyp
sin θ = _ adj hyp
cos θ = _ opp adj
tan θ = _
Notice that the cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent ratios, respectively. These are called the reciprocal functions. 1 csc θ = _
1 sec θ = _
sin θ
1 cot θ = _
cos θ
tan θ
The domain of any trigonometric function is the set of all acute angles θ of a right triangle. So, trigonometric functions depend only on the measures of the acute angles, not on the side lengths of a right triangle.
Example 2 Find Trigonometric Ratios 5 If sin B = _ , find the exact values of the five remaining trigonometric 8
functions for B. Step 1 Draw a right triangle and label one acute angle B. Label the opposite side 5 and the hypotenuse 8.
a
#
Step 2 Use the Pythagorean Theorem to find a. a2 + b2 = c2
Pythagorean Theorem
a2
b = 5 and c = 8
+
52
=
82
a2 + 25 = 64 a2
$ 5
8
"
Simplify.
= 39
Subtract 25 from each side.
a = ± √ 39
Take the square root of each side.
a = √ 39
Length cannot be negative.
Step 3 Find the other values. hyp
5 _8 Since sin B = _ , csc B = _ opp or . 8 adj √ 39 cos B = _ = _ 8 hyp
5
hyp adj
Labeling Triangles Throughout this chapter, a capital letter is used to represent both a vertex of a triangle and the measure of the angle at that vertex. The same letter in lowercase is used to represent both the side opposite that angle and the length of the side.
√ 39
adj √ 39 _ cot B = _ opp = 5
opp 5 √ 39 5 tan B = _ = _ or _ 39 adj √ 39
ReadingMath
8 √ 39 39
8 sec B = _ = _ or _
GuidedPractice 3 2. If tan B = _ , find exact values of the remaining trigonometric fuctions for B. 7
Angles that measure 30°, 45°, and 60° occur frequently in trigonometry.
KeyConcept Trigonometric Values for Special Angles 30°60°90° 1 sin 30° = _
2 √ 3 sin 60° = _ 2
√ 3 2 1 cos 60° = _ 2
cos 30° = _
√ 3 3
tan 30° = _
x
60°
2x 30°
tan 60° = √ 3 x √⎯ 3
45°45°90° √ 2 2
sin 45° = _
√ 2 2
cos 45° = _
tan 45° = 1
45°
x √⎯ 2
x
x
45°
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2
Use Trigonometric Functions You can use trigonometric functions to find missing side lengths and missing angle measures of right triangles.
Example 3 Find a Missing Side Length
StudyTip Choose a Function If the length of the hypotenuse is unknown, then either the sine or cosine function must be used to find the missing measure.
Use a trigonometric function to find the value of x. Round to the nearest tenth if necessary.
8
The length of the hypotenuse is 8. The missing measure is for the side adjacent to the 30° angle. Use the cosine function to find x. adj hyp _ cos 30° = x 8 √ 3 _ = _x 2 8 √ 8 3 _=x 2
cos θ = _
6.9 ≈ x
x
30°
Cosine function Replace θ with 30°, adj with x, and hyp with 8. cos 30° =
√ 3 _ 2
Multiply each side by 8. Use a calculator.
GuidedPractice 3A.
3B. x
60°
10
14
45°
x
You can use a calculator to find the missing side lengths of triangles that do not have 30°, 45°, or 60° angles.
Example 4 Find a Missing Side Length BUILDINGS To calculate the height of a building, Joel walked 200 feet from the base of the building and used an inclinometer to measure the angle from his eye to the top of the building. If his eye level is at 6 feet, how tall is the building?
RealWorldLink Inclinometers measure the angle of Earth’s magnetic field as well as the pitch and roll of vehicles, sailboats, and airplanes. They are also used for monitoring volcanoes and well drilling. Source: Science Magazine
d
The measured angle is 76°. The side adjacent to the angle is 200 feet. The missing measure is the side opposite the angle. Use the tangent function to find d. opp adj _ tan 76° = d 200
tan θ = _
200 tan 76° = d 802 ≈ d
76°
Tangent function
200 ft
Replace θ with 76°, opp with d, and adj with 200. Multiply each side by 200. Use a calculator to simplify: 200
76
.
Because the inclinometer was 6 feet above the ground, the height of the building is approximately 808 feet.
GuidedPractice 4. Use a trigonometric function to find the value of x. Round to the nearest tenth if necessary.
810  Lesson 131  Trigonometric Functions in Right Triangles
24° x
11
When solving equations like 3x = 27, you use the inverse of multiplication to find x. You also can find angle measures by using the inverse of sine, cosine, or tangent.
ReadingMath Inverse Trigonometric Ratios The expression sin1 x is read the inverse sine of x and is interpreted as the angle whose sine is x. Be careful not to confuse this notation with the notation for negative exponents; 1 sin1 x ≠ _ . Instead,
KeyConcept Inverse Trigonometric Ratios Words
If ∠A is an acute angle and the sine of A is x, then the inverse sine of x is the measure of ∠A.
Symbols
If sin A = x, then sin1 x = m∠A.
Example
1 1 sin A = _ → sin1 _ = m∠A → m∠A = 30°
Words
If ∠A is an acute angle and the cosine of A is x, then the inverse cosine of x is the measure of ∠A.
Symbols
If cos A = x, then cos1 x = m∠A.
Example
cos A = _ → cos1 _ = m∠A → m∠A = 45°
Words
If ∠A is an acute angle and the tangent of A is x, then the inverse tangent of x is the measure of ∠A.
Symbols
If tan A = x, then tan1 x = m∠A.
Example
tan A = √ 3 → tan1 √ 3 = m∠A → m∠A = 60°
2
sin x
this notation is similar to the notation for an inverse function, f1(x).
2
√ 2 2
√2 2
If you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle, which is the inverse of the trigonometric ratio.
Example 5 Find a Missing Angle Measure Find the measure of each angle. Round to the nearest tenth if necessary. a. ∠N .
/
6
You know the measure of the side opposite ∠N and the measure of the hypotenuse. Use the sine function. 6 sin N = _
10
10
sin1 
6 _ = m∠N 10
36.9° ≈ m∠N
sin θ =
opp _ hyp
Inverse sine Use a calculator.
b. ∠B "
Use the cosine function. 8 cos B = _ 16
8 cos1 _ = m∠B
16
16
60° = m∠B
$
8
cos θ =
adj _ hyp
Inverse cosine Use a calculator.
#
GuidedPractice Find x. Round to the nearest tenth if necessary. 5A.
5B. 17
18 x°
27
x°
15
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StudyTip Angles of Elevation and Depression The angle of elevation and the angle of depression are congruent since they are alternate interior angles of parallel lines.
In the figure at the right, the angle formed by the line of sight from the swimmer and a line parallel to the horizon is called the angle of elevation. The angle formed by the line of sight from the lifeguard and a line parallel to the horizon is called the angle of depression.
Angle of depression
Angle of elevation
Example 6 Use Angles of Evelation and Depression a. GOLF A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12°, find the distance from the tee to the hole.
36 yd
12°
Write an equation using a trigonometric function that involves the ratio of the vertical rise (side opposite the 12° angle) and the distance from the tee to the hole (hypotenuse). 36 sin 12° = _ x
x sin 12° = 36
sin θ =
opp _ hyp
Multiply each side by x.
36 x=_
Divide each side by sin 12°.
x ≈ 173.2
Use a calculator.
sin 12°
So, the distance from the tee to the hole is about 173.2 yards. b. ROLLER COASTER The hill of the roller coaster has an angle of descent, or an angle of depression, of 60°. Its vertical drop is 195 feet. Estimate the length of the hill.
60° x
195 ft
Write an equation using a trigonometric function that involves the ratio of the vertical drop (side opposite the 60° angle) and the length of the hill (hypotenuse).
RealWorldLink The steepest roller coasters in the world have angles of descent that are close to 90°. Source: Ultimate Roller Coaster
195 sin 60° = _ x
x sin 60° = 195
sin θ =
opp _ hyp
Multiply each side by x.
195 x=_
Divide each side by sin 60°.
x ≈ 225.2
Use a calculator.
sin 60°
So, the length of the hill is about 225.2 feet.
GuidedPractice 6A. MOVING A ramp for unloading a moving truck has an angle of elevation of 32°. If the top of the ramp is 4 feet above the ground, estimate the length of the ramp. 6B. LADDERS A 14ft long ladder is placed against a house at an angle of elevation of 72°. How high above the ground is the top of the ladder?
812  Lesson 131  Trigonometric Functions in Right Triangles
Check Your Understanding Example 1
Find the values of the six trigonometric functions for angle θ. 1.
2.
θ
16
6
8
Example 2
= StepbyStep Solutions begin on page R20.
θ 12
In a right triangle, ∠A is acute. Find the values of the five remaining trigonometric funtions. 4 3. If cos A = _ , what is sin A?
20 4. If tan A = _ , what is cos A?
7
21
Examples 3–4 Use a trigonometric function to find the value of x. Round to the nearest tenth. 5
6.
60°
52°
x
33°
x
Find the value of x. Round to the nearest tenth. 15
8. 8
Example 6
7
6 x
22
Example 5
7.
9. 6
10.
14
16
x°
6
x°
x°
11. GEOGRAPHY Christian found two trees directly across from each other in a canyon. When he moved 100 feet from the tree on his side (parallel to the edge of the canyon), the angle formed by the tree on his side, Christian, and the tree on the other side was 70°. Find the distance across the canyon. 12. LADDERS The recommended angle of elevation for a ladder used in fire fighting is 75°. At what height on a building does a 21foot ladder reach if the recommended angle of elevation is used? Round to the nearest tenth.
Practice and Problem Solving Example 1
Extra Practice begins on page 947.
Find the values of the six trigonometric functions for angle θ. 12
13. θ
14.
9 40
13 θ
15. 10
θ
16. 7
9
6
θ
Example 2
In a right triangle, ∠A and ∠B are acute. Find the values of the five remaining trigonometric funtions. 8 17. If tan A = _ , what is cos A?
3 18. If cos A = _ , what is tan A?
19. If tan B = 3, what is sin B?
4 20. If sin B = _ , what is tan B? 9
15
10
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Examples 3–4 Use a trigonometric function to find each value of x. Round to the nearest tenth. 9
21.
22. 4
x
23
64°
x 30° 18
x
45°
24.
25. 22
x
x
48°
26. 32°
15
70°
x
14
27. PARASAILING Refer to the beginning of the lesson and the figure at the right. Find a, the altitude of a person parasailing, if the tow rope is 250 feet long and the angle formed is 32°. Round to the nearest tenth.
250 ft
a
32°
28. BRIDGES Devon wants to build a rope bridge between his treehouse and Cheng’s treehouse. Suppose Devon’s treehouse is directly behind Cheng’s treehouse. At a distance of 20 meters to the left of Devon’s treehouse, an angle of 52° is measured between the two treehouses. Find the length of the rope. Example 5
Find the value of x. Round to the nearest tenth. 29.
30.
x°
31. 19
5
10
8 9
x°
x° 12
32.
33.
34.
4
x°
x°
32
7
x°
27
25
10
Example 6
35. SQUIRRELS Adult flying squirrels can make glides of up to 160 feet. If a flying squirrel glides a horizontal distance of 160 feet and the angle of descent is 9°, find its change in height. 36. HANG GLIDING A hang glider climbs at a 20° angle of elevation. Find the change in altitude of the hang glider when it has flown a horizontal distance of 60 feet.
B
Use trigonometric functions to find the values of x and y. Round to the nearest tenth. 37.
y
38.
39.
50 ° 30.2 46.5°
x
x° 71.8
x
y° 81
y
Solve each equation. 3 40. cos A = _
9 41. sin N = _
42. tan X = 15
43. sin T = 0.35
44. tan G = 0.125
45. cos Z = 0.98
19
11
814  Lesson 131  Trigonometric Functions in Right Triangles
26 3 4
46. MONUMENTS A monument casts a shadow 24 feet long. The angle of elevation from the end of the shadow to the top of the monument is 50°. a. Draw and label a right triangle to represent this situation. b. Write a trigonometric function that can be used to find the height of the monument. c. Find the value of the function to determine the height of the monument to the nearest tenth.
C
47 NESTS Tabitha’s eyes are 5 feet above the ground as she looks up to a bird’s nest in a tree. If the angle of elevation is 74.5° and she is standing 12 feet from the tree’s base, what is the height of the bird’s nest? Round to the nearest foot. 48. RAMPS Two bicycle ramps each cover a horizontal distance of 8 feet. One ramp has a 20° angle of elevation, and the other ramp has a 35° angle of elevation, as shown at the right.
20°
a. How much taller is the second ramp than the first? Round to the nearest tenth. b. How much longer is the second ramp than the first? Round to the nearest tenth.
35° 8 ft
49. FALCONS A falcon at a height of 200 feet sees two mice A and B, as shown in the diagram. a. What is the approximate distance z between the falcon and mouse B?
62° 200 ft
[ 72° "
Y
#
Z
b. How far apart are the two mice? In ABC, ∠C is a right angle. Use the given measurements to find the missing side lengths and missing angle measures of ABC. Round to the nearest tenth if necessary. 50. m∠A = 36°, a = 12
51. m∠B = 31°, b = 19
52. a = 8, c = 17
4 53. tan A = _ ,a=6
H.O.T. Problems
5
Use HigherOrder Thinking Skills y
54. CHALLENGE A line segment has endpoints A(2, 0) and B(6, 5), as shown in the figure at the right. What is the measure of the acute angle θ formed by the line segment and the xaxis? Explain how you found the measure.
#
55. REASONING Determine whether the following statement is true or false. Explain your reasoning. For any acute angle, the sine function will never have a negative value.
"
θ x
0
56. OPEN ENDED In right triangle ABC, sin A = sin C. What can you conclude about ABC? Justify your reasoning. 57.
2 WRITING IN MATH A roof has a slope of _ . Describe the connection between the 3 slope and the angle of elevation θ that the roof makes with the horizontal. Then use an inverse trigonometric function to find θ.
E
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SPI 3103.3.13, SPI 3108.4.7
Standardized Test Practice 60. A hot dog stand charges price x for a hot dog and price y for a drink. Two hot dogs and one drink cost $4.50. Three hot dogs and two drinks cost $7.25. Which matrix could be multiplied by ⎡ 4.50 ⎤ to find x and y? ⎢ ⎣ 7.25 ⎦
58. EXTENDED RESPONSE Your school needs 5 cases of yearbooks. Neighborhood Yearbooks lists a case of yearbooks at $153.85 with a 10% discount on an order of 5 cases. Yearbooks R Us lists a case of yearbooks at $157.36 with a 15% discount on 5 cases. a. Which company would you choose?
⎡ 1 ⎣ 2
1⎤ 1 ⎦
C ⎢
⎡ 2 ⎣ 3
1 ⎤ 2⎦
D ⎢
A ⎢
b. What is the least amount that you would have to spend for the yearbooks?
B ⎢
59. SHORT RESPONSE As a fundraiser, the marching band sold Tshirts and hats. They sold a total of 105 items and raised $1170. If the cost of a hat was $10 and the cost of a Tshirt was $15, how many Tshirts were sold?
⎡ 1 ⎣ 1
2⎤ 3⎦
⎡ 1 ⎣ 1
1 ⎤ 2⎦
61. SAT/ACT The length and width of a rectangle are in the ratio of 5:12. If the rectangle has an area of 240 square centimeters, what is the length, in centimeters, of its diagonal? F 24
H 28
G 26
J 30
K 32
Spiral Review 62. POLLS A polling company wants to estimate how many people are in favor of a new environmental law. The polling company polls 20 people. The probability that a person is in favor of the law is 0.5. (Lesson 127) a. What is the probability that exactly 12 people are in favor of the new law? b. What is the expected number of people in favor of the law? Text each null hypothesis. Write accept or reject. (Lesson 126) 63. H0 = 92, H1 > 92, n = 80, x− = 92.75, and s = 2.8 64. H = 48, H > 48, n = 240, x− = 48.2, and s = 2.2 0
1
65. H0 = 71, H1 > 71, n = 180, x− = 72.4, and s = 3.5 66. H = 55, H < 55, n = 300, x− = 54.5, and s = 1.9 0
1
Find each probability. (Lesson 123) 67. A city council consists of six Democrats, two of whom are women, and six Republicans, four of whom are men. A member is chosen at random. If the member chosen is a man, what is the probability that he is a Democrat? 68. Two boys and two girls are lined up at random. What is the probability that the girls are separated if a girl is on an end?
Skills Review Find each product. Include the appropriate units with your answer. (Extend Lesson 61)
(
)
5280 feet 69. 4.3 miles _
(
1 mile
)
18 cubic inches 72. __ 24 seconds 5 seconds
(
8 pints 70. 8 gallons _ 1 gallon
)
10 centimeters 73. 65 degrees __
(
3 degrees
816  Lesson 131  Trigonometric Functions in Right Triangles
( 3 meters ) 7 liters 74. (_ 10 minutes 30 minutes ) 5 dollars 71. _ 21 meters
)
Angles and Angle Measure Then
Now
Why?
You used angles with degree measures.
1 2
A sundial is an instrument that indicates the time of day by the shadow that it casts on a surface marked to show hours or fractions of hours. The shadow moves around the dial 15° every hour.
(Lesson 131)
NewVocabulary standard position initial side terminal side coterminal angles radian central angle arc length
Draw and find angles in standard position. Convert between degree measures and radian measures.
1
90° y
Angles in Standard Position An angle on the
coordinate plane is in standard position if the vertex is at the origin and one ray is on the positive xaxis. • The ray on the xaxis is called the initial side of the angle.
180° terminal side
• The ray that rotates about the center is called the terminal side.
vertex 0 initial side
x
0°
270°
KeyConcept Angle Measures Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems. ✔ 3103.4.1 Convert between radians and degrees and vice versa. ✔ 3103.4.4 Understand the relationship between the radius, the central angle, and radian measure.
y
If the measure of an angle is positive, the terminal side is rotated counterclockwise.
y 120°
If the measure of an angle is negative, the terminal side is rotated clockwise.
x
0
0 145°
x
Example 1 Draw an Angle in Standard Position Draw an angle with the given measure in standard position. a. 215° 215° = 180° + 35° b. 40° Draw the terminal side of the angle 35° counterclockwise past the negative xaxis. y
The angle is negative. Draw the terminal side of the angle 40° clockwise from the positive xaxis. y
215° 35°
0
x
0
40°
x
GuidedPractice 1A. 80°
1B. 105°
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The terminal side of an angle can make more than one complete rotation. For example, a complete rotation of 360° plus a rotation of 120° forms an angle that measures 360° + 120° or 480°.
480° angle y 120° x
0 360°
RealWorld Example 2 Draw an Angle in Standard Position WAKEBOARDING Wakeboarding is a combination of surfing, skateboarding, snowboarding, and water skiing. One maneuver involves a 540degree rotation in the air. Draw an angle in standard position that measures 540°. y
540° = 360° + 180° Draw the terminal side of the angle 180° past the positive xaxis.
540° x
0
RealWorldLink
GuidedPractice
Wakeboarding is one of the fastestgrowing water sports in the United States. Participation increased more than 100% in recent years.
2. Draw an angle in standard position that measures 600°.
Source: King of Wake
Two or more angles in standard position with the same terminal side are called coterminal angles. For example, angles that measure 60°, 420°, and 300° are coterminal, as shown in the figure at the right. An angle that is coterminal with another angle can be found by adding or subtracting a multiple of 360°.
y 60°
420°
x
0 300°
• 60° + 360° = 420° • 60°  360° = 300°
ReadingMath
Example 3 Find Coterminal Angles
Angle of Rotation In trigonometry, an angle is sometimes referred to as an angle of rotation.
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. a. 130° positive angle: 130° + 360° = 490° negative angle: 130°  360° = 230°
Add 360°. Subtract 360°.
b. 200° positive angle: 200° + 360° = 160° negative angle: 200°  360° = 560°
Add 360°. Subtract 360°.
GuidedPractice 3A. 15°
818  Lesson 132  Angles and Angle Measure
3B. 45°
StudyTip Radians As with degrees, radians measure the amount of rotation from the initial side to the terminal side. • The measure of an angle in radians is positive if its rotation is counterclockwise. • The measure is negative if the rotation is clockwise.
2
Convert Between Degrees and Radians Angles can
y
also be measured in units that are based on arc length. One radian is the measure of an angle θ in standard position with a terminal side that intercepts an arc with the same length as the radius of the circle.
r
x
0
The circumference of a circle is 2πr. So, one complete revolution around a circle equals 2π radians. Since 2π radians = 360°, degree measure and radian measure are related by the following equations. 2π radians = 360°
r
θ
θ = 1 radian
π radians = 180°
KeyConcept Convert Between Degrees and Radians Degrees to Radians
Radians to Degrees
To convert from degrees to radians, multiply the number of degrees by
To convert from radians to degrees, multiply the number of radians by
π radians _ .
180° _ . π radians
180°
ReadingMath Radian Measures The word radian is usually omitted when angles are expressed in radian measure. Thus, when no units are given for an angle measure, radian measure is implied.
Example 4 Convert Between Degrees and Radians Rewrite the degree measure in radians and the radian measure in degrees.
_
b. 5π
a. 30°
2
π radians 30° = 30° · _ 180°
5π 5π 180° _ =_ radians · _ 2
30π π =_ or _ radians 180
2
π radians
900° =_ or 450° 2
6
GuidedPractice 3π 4B. _
4A. 120°
8
ConceptSummary Degrees and Radians y
The diagram shows equivalent degree and radian measures for special angles. You may find it helpful to memorize the following equivalent degree and radian measures. The other special angles are multiples of these angles. π 30° = _ 6 π 60° = _ 3
π 45° = _ 4 π 90° = _ 2
5π 6
3π 4
π 2
2π 3
120° 135° 150°
7π 6 5π 4
π 3
π 4
60° 45° 30°
π 6
0°
180°
π
90°
0 2π x ° 360
0
330° 315° 300°
210° 225° 240° 4π 3
270° 3π 2
5π 3
11π 6
7π 4
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A central angle of a circle is an angle with a vertex at the center of the circle. If you know the measure of a central angle and the radius of the circle, you can find the length of the arc that is intercepted by the angle.
intercepted arc central angle θ
Corollaries Words
For a circle with radius r and central angle θ (in radians), the arc length s equals the product of r and θ.
Model θ
s
r
Symbols
s = rθ You will justify this formula in Exercise 52
RealWorld Example 5 Find Arc Length TRUCKS Monster truck tires have a radius of 33 inches. How far does a monster truck travel in feet after just three fourths of a tire rotation? Step 1 Find the central angle in radians. 3 3π θ=_ · 2π or _
WatchOut! Arc Length Remember to write the angle measure in radians, not degrees, when finding arc length. Also, recall that the number of radians in a complete rotation is 2π.
2
4
_
The angle is 3 of a complete rotation. 4
Step 2 Use the radius and central angle to find the arc length. s = rθ
Write the formula for arc length.
3π = 33 · _
3π Replace r with 33 and θ with _ .
≈ 155.5 in.
Use a calculator to simplify.
≈ 13.0 ft
Divide by 12 to convert to feet.
2
2
So, the truck travels about 13 feet after three fourths of a tire rotation.
GuidedPractice 5. A circle has a diameter of 9 centimeters. Find the arc length if the central angle is 60°. Round to the nearest tenth.
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Draw an angle with the given measure in standard position. 1. 140° Example 3
5 175°
6. 100°
Rewrite each degree measure in radians and each radian measure in degrees. π 7. _ 4
Example 5
3. 390°
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. 4. 25°
Example 4
2. 60°
8. 225°
9. 40°
10. TENNIS A tennis player’s swing moves along the path of an arc. If the radius of the arc’s circle is 4 feet and the angle of rotation is 100°, what is the length of the arc? Round to the nearest tenth.
820  Lesson 132  Angles and Angle Measure
Practice and Problem Solving
Extra Practice begins on page 947
Examples 1–2 Draw an angle with the given measure in standard position. 11. 75°
12. 160°
13. 90°
14. 120°
15. 295°
16. 510°
17. GYMNASTICS A gymnast on the uneven bars swings to make a 240° angle of rotation. 18. FOOD The lid on a jar of pasta sauce is turned 420° before it comes off. Example 3
Example 4
Example 5
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. 19. 50°
20. 95°
21. 205°
22. 350°
23. 80°
24. 195°
Rewrite each degree measure in radians and each radian measure in degrees. 25 330°
5π 26. _
π 27. _
28. 50°
29. 190°
7π 30. _
6
3
3
31. SKATEBOARDING The skateboard ramp at the right is called a quarter pipe. The curved surface is determined by the radius of a circle. Find the length of the curved part of the ramp.
" 8 ft
32. RIVERBOATS The paddlewheel of a riverboat has a diameter of 24 feet. Find the arc length of the circle made when the paddlewheel rotates 300°.
#
Find the length of each arc. Round to the nearest tenth. 33.
34. 5 cm
3π 7
10π 9
27 m
B
35. CLOCKS How long does it take for the minute hand on a clock to pass through 2.5π radians? 36. SUNDIALS Refer to the beginning of the lesson. A shadow moves around a sundial 15° every hour. 8π a. After how many hours is the angle of rotation of the shadow _ radians? 5
b. What is the angle of rotation in radians after 5 hours? c. A sundial has a radius of 8 inches. What is the arc formed by a shadow after 14 hours? Round to the nearest tenth. Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. 37. 620°
38. 400°
3π 39. _ 4
19π 40. _ 6
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41 SWINGS A swing has a 165° angle of rotation. a. Draw the angle in standard position. b. Write the angle measure in radians. 1 c. If the chains of the swing are 6_ feet long, what is the length of the arc that the 2 swing makes? Round to the nearest tenth.
d. Describe how the arc length would change if the lengths of the chains of the swing were doubled. MULTIPLE REPRESENTATIONS Consider A(4, 0), B(4, 6), C(6, 0), and D(6, 8).
42.
a. Geometric Draw EAB and ECD with E at the origin. b. Algebraic Find the values of the tangent of ∠BEA and the tangent of ∠DEC. −− −− c. Algebraic Find the slope of BE and ED. d. Verbal What conclusions can you make about the relationship between slope and tangent? Rewrite each degree measure in radians and each radian measure in degrees. 21π 43. _
45. 200°
44. 124°
8
46. 5
47. CAROUSELS A carousel makes 5 revolutions per minute. The circle formed by riders sitting in the outside row has a radius of 17.2 feet. The circle formed by riders sitting in the inside row has a radius of 13.1 feet.
17.2 ft
a. Find the angle θ in radians through which the carousel rotates in one second.
13.1 ft
b. In one second, what is the difference in arc lengths between the riders sitting in the outside row and the riders sitting in the inside row?
H.O.T. Problems
Use HigherOrder Thinking Skills y
48. ERROR ANALYSIS Tarshia and Alan are writing an expression for the measure of an angle coterminal with the angle shown at the right. Is either of them correct? Explain your reasoning. 0
Tarshia The measure of a coterminal angle is (x – 360)°.
Alan
x°
x
The measure of a coterminal angle is (360 – x)°.
π 49. CHALLENGE A line makes an angle of _ radians with the positive xaxis at the 2 point (2, 0). Find an equation for this line. 1 50. REASONING Express _ of a revolution in degrees and in radians. Explain 8 your reasoning.
51. OPEN ENDED Draw and label an acute angle in standard position. Find two angles, one positive and one negative, that are coterminal with the angle. 52. REASONING Justify the formula for the length of an arc. 53. WRITING IN MATH Use a circle with radius r to describe what one degree and one radian represent. Then explain how to convert between the measures.
822  Lesson 132  Angles and Angle Measure
SPI 3103.3.2, SPI 3108.4.14, SPI 3103.3.4
Standardized Test Practice 54. SHORT RESPONSE If (x + 6)(x + 8) (x  7)(x  5) = 0, find x.
56. GEOMETRY If the area of the figure is 60 square −− units, what is the length of side XZ? ;
55. Which of the following represents an inverse variation? A
B
C
D
x
2
5
10
20
25
50
y
50
20
10
5
4
2
x
2
4
6
8
10
12
y
4
8
12
16
20
24
x
1
2
3
4
5
6
y
5
10
15
20
25
30
x
10
9
8
7
6
5
y
5
6
7
8
9
10
6
:
9
F 2 √ 34
H 4 √ 109
G 2 √ 109
J
4 √ 34
57. SAT/ACT The first term of a sequence is 6, and every term after the first is 8 more than the term immediately preceding it. What is the value of the 101st term? A 788 B 794 C 802
D 806 E 814
Spiral Review Find the values of the six trigonometric functions for angle θ. (Lesson 131) 58.
59.
3 14
θ
60. 22
13
11
θ
θ 15
A binomial distribution has a 40% rate of success. There are 12 trials. (Lesson 127) 61. What is the probability that there will be at least 8 successes? exactly 5 failures? 62. What is the expected number of successes? 63. MANUFACTURING The sizes of CDs made by a company are normally distributed with a standard deviation of 1 millimeter. The CDs are supposed to be 120 millimeters in diameter, and they are made for drives that are 122 millimeters wide. (Lesson 125) a. What percent of the CDs would you expect to be greater than 120 millimeters? b. If the company manufactures 1000 CDs per hour, how many of the CDs made in one hour would you expect to be between 119 and 122 millimeters? c. About how many CDs per hour will be too large to fit in the drives? 64. FINANCIAL LITERACY If the rate of inflation is 2%, the cost of an item in future years can be found by iterating the function c(x) = 1.02x. Find the cost of a $70 digital audio player in four years if the rate of inflation remains constant. (Lesson 115)
Skills Review Use the Pythagorean Theorem to find the length of the hypotenuse for each right triangle with the given side lengths. (Lesson 07) 65. a = 12, b = 15
66. a = 8, b = 17
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Geometry Lab
Areas of Parallelograms Tennessee Curriculum Standards
The area of any triangle can be found using the sine ratios in the triangle. A similar process can be used to find the area of a parallelogram.
CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
Activity Find the area of parallelogram ABCD.
#
$
16 in. 60° 28 in.
"
−− Step 1 Draw diagonal BD. −− BD divides the parallelogram into two congruent triangles, ABD and CDB.
1 Area = _ (AB)(AD) sin A 2
1 =_ (16)(28) sin 60° 2
(2)
#
$
16 in. 60° 28 in.
"
Step 2 Find the area of ABD.
%
%
Area of a triangle AB = 16, AD = 28, and A = 60°
√ 3 = 224 _
Multiply and evaluate sin 60°.
= 112 √ 3
Simplify.
Step 3 Find the area of ABCD. The area of ABCD is equal to the sum of the areas of ABD and CDB. Because ABD CDB, the areas of ABD and CDB are equal. So, the area of ABCD equals twice the area of ABD. 2 · 112 √ 3 = 224 √ 3 or about 387.98 square inches.
Exercises For each of the following, a. find the area of each parallelogram. b. find the area of each parallelogram when the included angle is half the given measure. c. find the area of each parallelogram when the included angle is twice the given measure. 1.
10 m 45° 15 m
2.
5 in.
3.
30° 9 in.
824  Extend 132  Geometry Lab: Areas of Parallelograms
100 ft 75° 200 ft
Trigonometric Functions of General Angles Then
Now
Why?
You found values of trigonometric functions for acute angles. (Lesson 131)
1
Find values of trigonometric functions for general angles.
2
Find values of trigonometric functions by using reference angles.
In the ride at the right, the cars rotate back and forth about a central point. The positions of the arms supporting the cars can be described using trigonometric angles in standard position, with the central point of the ride at the origin of a coordinate plane.
NewVocabulary quadrantal angle reference angle
1
Trigonometric Functions for General Angles You can find values of trigonometric functions for angles greater than 90° or less than 0°.
KeyConcept Trigonometric Functions of General Angles Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
y
Let θ be an angle in standard position and let P(x, y) be a point on its terminal side. Using the Pythagorean Theorem, r = √ x2 + y2 . The six trigonometric functions of θ are defined below. y r
sin θ = _
cos θ = _x r
csc θ = _r , y ≠ 0 y
θ
1 (x, y )
sec θ = _r , x ≠ 0 cot θ = _x , y ≠ 0 x
x
0
y x
tan θ = _, x ≠ 0
r
y
Example 1 Evaluate Trigonometric Functions Given a Point y
The terminal side of θ in standard position contains the point at (3, 4). Find the exact values of the six trigonometric functions of θ.
θ
Step 1 Draw the angle, and find the value of r. r=
x
0
x2 + y2 √
r (3, 4)
= √ (3)2 + (4)2 = √ 25 or 5
Step 2 Use x = 3, y = 4, and r = 5 to write the six trigonometric ratios. y 5 r 5 5 r _ _ _ csc θ = = or  5 y 4 4 4 4 sin θ = _ = _ or _
3 x 3 cos θ = _ =_ or _ 5 r 5 5 5 r _ _ _ sec θ = = or 3 x 3
y 3 3 x 3 3 x cot θ = _ =_ or _ y 4 4
4 4 tan θ = _ = _ or _
GuidedPractice 1. The terminal side of θ in standard position contains the point at (6, 2). Find the exact values of the six trigonometric functions of θ.
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If the terminal side of angle θ in standard position lies on the x or yaxis, the angle is called a quadrantal angle.
StudyTip Quadrantal Angles The measure of a quadrantal angle is a multiple of
KeyConcept Quadrantal Angles θ = 0° or 0 radians
π 90° or _ . 2
θ = 90° π or _ radians 2
y
y (0, r ) (r, 0) x
θ = 270° 3π or _ radians 2
y
y
θ
θ
θ
θ 0
θ = 180° or π radians
x
0
x
(r, 0) 0
x
0 (0, r )
Example 2 Quadrantal Angles y
The terminal side of θ in standard position contains the point at (0, 6). Find the values of the six trigonometric functions of θ.
(0, 6)
y 6 r r _ _ csc θ = = 6 or 1 6 y
x
0
The point at (0, 6) lies on the positive yaxis, so the quadrantal angle θ is 90°. Use x = 0, y = 6, and r = 6 to write the trigonometric functions. 6 sin θ = _ = _ or 1
θ
y 0 x 0 x _ _ cot θ = = or 0 6 y
0 x cos θ = _ =_ or 0
6 tan θ = _ = _ undefined
6 r 6 r _ _ sec θ = = undefined 0 x
GuidedPractice 2. The terminal side of θ in standard position contains the point at (2, 0). Find the values of the six trigonometric functions of θ.
ReadingMath Theta Prime θ is read theta prime.
2
y
Trigonometric Functions with Reference Angles If θ is
a nonquadrantal angle in standard position, its reference angle θ is the acute angle formed by the terminal side of θ and the xxaxis. The rules for finding the measures of reference angles for 0° < θ < 360° or 0° < θ < 2π are shown below. fo
θ x
0
θ'
KeyConcept Reference Angles 2VBESBOU*
2VBESBOU**
2VBESBOU***
y
y
0
y
θ x
0
θ'
θ' x
θ = θ
θ'
θ
θ 0
θ
2VBESBOU*7
y
0
θ = 180°  θ θ = π  θ
826  Lesson 133  Trigonometric Functions of General Angles
x
θ = θ  180° θ = θ  π
θ = 360°  θ θ = 2π  θ
x
If the measure of θ is greater than 360° or less than 0°, then use a coterminal angle with a positive measure between 0° and 360° to find the reference angle.
Example 3 Find Reference Angles
StudyTip Graphing Angles You can refer to the diagram in the Lesson 132 Concept Summary to help you sketch angles.
Sketch each angle. Then find its reference angle.
_
b.  5π
a. 210°
4
3π 5π coterminal angle: _ + 2π = _
y
y
θ = 210° θ'
4
4
x
0
θ = 3π 4
θ'
x
0
The terminal side of 210° lies in Quadrant III. θ = θ  180° = 210°  180° or 30°
3π The terminal side of _ lies 4 in Quadrant III. θ = π  θ 3π π =π_ or _ 4
4
GuidedPractice 2π 3B. _
3A. 110°
3
You can use reference angles to evaluate trigonometric functions for any angle θ. The sign of a function is determined by the quadrant in which the terminal side of θ lies. Use these steps to evaluate a trigonometric function for any angle θ.
KeyConcept Evaluate Trigonometric Functions Step 1 Find the measure of the reference angle θ. Step 2 Evaluate the trigonometric function for θ. Step 3 Determine the sign of the trigonometric function value. Use the quadrant in which the terminal side of θ lies.
2VBESBOU**
2VBESBOU*
sin θ, csc θ: +
sin θ, csc θ: +
cos θ, sec θ: 
cos θ, sec θ: +
tan θ, cot θ: 
tan θ, cot θ: +
2VBESBOU***
2VBESBOU*7
sin θ, csc θ: 
sin θ, csc θ: 
cos θ, sec θ: 
cos θ, sec θ: +
tan θ, cot θ: +
tan θ, cot θ: 
You can use the trigonometric values of angles measuring 30°, 45°, and 60° that you learned in Lesson 131.
Sine 1 sin 30° = _ 2 √2 _ sin 45° = 2 √3 _ sin 60° = 2
Cosine
Trigonometric Values for Special Angles Tangent Cosecant
√ 3 cos 30° = _ 2 √ 2 cos 45° = _ 2
√ 3 tan 30° = _ 3
1 cos 60° = _ 2
Secant
Cotangent
csc 30° = 2
2 √ 3 sec 30° = _ 3
cot 30° = √ 3
tan 45° = 1
csc 45° = √ 2
sec 45° = √2
cot 45° = 1
tan 60° = √ 3
csc 60° = _
sec 60° = 2
cot 60° = _
2 √ 3 3
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√ 3 3
827
Example 4 Use a Reference Angle to Find a Trigonometric Value Find the exact value of each trigonometric function. a. cos 240° y
The terminal side of 240° lies in Quadrant III. θ = θ  180° = 240°  180° or 60°
Find the measure of the reference angle. θ = 240°
1 cos 240° = cos 60° or _ 2
The cosine function is negative in Quadrant III.
θ = 240° x
0 θ' = 60°
_
b. csc 5π 6
5π The terminal side of _ lies in Quadrant II.
y
6
θ = π  θ 6
6
5π π csc _ = csc _ 6
θ = 5π
Find the measure of the reference angle.
5π π =π_ or _ 6
= csc 30° =2
5π θ=_
6
θ' = π 6
6
x
0
The cosecant function is positive in Quadrant II.
_π radians = 30° 6 1 csc 30° = _ sin 30
GuidedPractice 5π 4B. tan _
4A. cos 135°
6
RealWorld Example 5 Use Trigonometric Functions RIDES The swing arms of the ride at the right are 84 feet long and the height of the axis from which the arms swing is 97 feet. What is the total height of the ride at the peak of the arc? coterminal angle: 200° + 360° = 160°
84 ft y ft 200°
x
97 ft
reference angle: 180°  160° = 20° y r y _ sin 20° = 84
sin θ = _
RealWorldLink On a swing ride, riders experience weightlessness just like the drop side of a roller coaster. The ride lasts one minute and reaches speeds of 60 miles per hour in both directions. Source: Cedar Point
84 sin 20° = y 28.7 ≈ y
Sine function θ = 20° and r = 84 Multiply each side by 84. Use a calculator to solve for y.
Since y is approximately 28.7 feet, the total height of the ride at its peak is 28.7 + 97 or about 125.7 feet.
GuidedPractice 5. RIDES A similar ride that is smaller has swing arms that are 72 feet long. The height of the axis from which the arms swing is 88 feet, and the angle of rotation from the standard position is 195°. What is the total height of the ride at the peak of the arc?
828  Lesson 133  Trigonometric Functions of General Angles
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ. 2. (8, 15)
1. (1, 2) Example 3
Sketch each angle. Then find its reference angle. 4. 300°
Example 4
3π 6. _
5. 115°
4
Find the exact value of each trigonometric function. 3π 7. sin _ 4
Example 5
3. (0, 4)
5π 8. tan _
9. sec 120°
3
11. ENTERTAINMENT Alejandra opens her portable DVD player so that it forms a
10. sin 300°
1
5 2 in.
1 125° angle. The screen is 5_ inches long.
125°
2
d
a. Redraw the diagram so that the angle is in standard position on the coordinate plane.
b. Find the reference angle. Then write a trigonometric function that can be used to find the distance to the wall d that she can place the DVD player. c. Use the function to find the distance. Round to the nearest tenth.
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ.
Example 3
12. (5, 12)
13 (6, 8)
14. (3, 0)
15. (0, 7)
16. (4, 2)
17. (9, 3)
Sketch each angle. Then find its reference angle. 18. 195°
19. 285°
20. 250°
7π 21. _
π 22. _
23. 400°
4
Example 4
4
Find the exact value of each trigonometric function. 24. sin 210°
25. tan 315°
26. cos 150°
27. csc 225°
4π 28. sin _
5π 29. cos _
5π 30. cot _
11π 31. sec _
3
Example 5
3
32. SOCCER A soccer player x feet from the goalie kicks the ball toward the goal, as shown in the figure. The goalie jumps up and catches the ball 7 feet in the air.
6
4
154°
7 ft
a. Find the reference angle. Then write a trigonometric function that can be used to find how far from the goalie the soccer player was when he kicked the ball.
x
b. About how far away from the goalie was the soccer player? connectED.mcgrawhill.com
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B
33 SPRINKLER A sprinkler rotating back and forth shoots water out a distance of 10 feet. From the horizontal position, it rotates 145° before reversing its direction. At a 145° angle, about how far to the left of the sprinkler does the water reach?
v
2
145°
10 ft
10 ft
sin 2θ 32
0 34. BASKETBALL The formula R = _ gives the distance of a basketball shot with an
initial velocity of V0 feet per second at an angle θ with the ground. a. If the basketball was shot with an initial velocity of 24 feet per second at an angle of 75°, how far will the basketball travel? b. If the basketball was shot at an angle of 65° and traveled 10 feet, what was its initial velocity? c. If the basketball was shot with an initial velocity of 30 feet per second and traveled 12 feet, at what angle was it shot? 35. PHYSICS A rock is shot off the edge of a ravine with a slingshot at an angle of 65° and with an initial velocity of 6 meters per second. The equation that represents the horizontal distance of the rock x is x = v0 (cos θ)t, where v0 is the initial velocity, θ is the angle at which it is shot, and t is the time in seconds. About how far does the rock travel after 4 seconds? y
36. FERRIS WHEELS The Wonder Wheel Ferris wheel at Coney Island has a radius of about 68 feet and is 15 feet off the ground. After a person gets on the bottom car, the Ferris wheel rotates 202.5° counterclockwise before stopping. How high above the ground is this car when it has stopped?
A
202.5°
68 ft x
? ft
C
Suppose θ is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of θ. 4 37. sin θ = _ , Quadrant II
2 38. tan θ = _ , Quadrant IV
8 39. cos θ = _ , Quadrant III 17
12 40. cot θ = _ , Quadrant IV 5
5
3
Find the exact value of each trigonometric function. 41. cot 270° 7π 44. tan _
(
H.O.T. Problems
6
42. csc 180°
)
43. sin 570°
11π 45. cos _
(
6
)
9π 46. cot _ 4
Use HigherOrder Thinking Skills √ 2
47. CHALLENGE For an angle θ in standard position, sin θ = _ and tan θ = 1. Can the 2 value of θ be 225°? Justify your reasoning. 48. REASONING Determine whether 3 sin 60° = sin 180° is true or false. Explain your reasoning. 49. REASONING Use the sine and cosine functions to explain why cot 180° is undefined. 50. OPEN ENDED Give an example of a negative angle θ for which sin θ > 0 and cos θ < 0. 51. WRITING IN MATH Describe the steps for evaluating a trigonometric function for an angle θ that is greater than 90°. Include a description of a reference angle.
830  Lesson 133  Trigonometric Functions of General Angles
SPI 3103.2.2
Standardized Test Practice 54. The expression (6 + i)2 is equivalent to which of the following expressions?
52. GRIDDED RESPONSE If the sum of two numbers is 21 and their difference is 3, what is their product? −− 53. GEOMETRY D is the midpoint of BC, and A and −−− −− E are the midpoints of BD and DC, respectively. −− If the length of AE is 12, what is the length −− of BC? A 6 B 12
F 12i G 36  i
H 36  12i J 35  12i
55. SAT/ACT Of the following, which is least? 1 A 1+_
1 D 1×_
4 _ B 1 1 4 _ C 1÷ 1 4
C 24 D 48
4
1 E _ 1 4
Spiral Review Rewrite each radian measure in degrees. (Lesson 132) 4 56. _ π
11 57. _ π
3
17 58. _ π
6
4
Solve each equation. (Lesson 131) 13 59. cos a = _
b 60. sin 30 = _
9 61. tan c = _
6
17
4
62. ARCHITECTURE A memorial being constructed in a city park will be a brick wall, with a top row of six goldplated bricks engraved with the names of six local war veterans. Each row has two more bricks than the row above it. Prove that the number of bricks in the top n rows is n2 + 5n. (Lesson 117) 63. LEGENDS There is a legend of a king who wanted to reward a boy for a good deed. The king gave the boy a choice. He could have $1,000,000 at once, or he could be rewarded daily for a 30day month, with one penny on the first day, two pennies on the second day, and so on, receiving twice as many pennies each day as the previous day. How much would the second option be worth? (Lesson 113) Write an equation for each circle given the endpoints of a diameter. (Lesson 103) 64. (2, 4), (10, 2)
65. (1, 10), (7, 6)
66. (9, 0), (4, 7)
6 3x 68. __  __ 2 2
2x 4 69. _ + __ 2 2
Simplify each expression. (Lesson 92) 5 x 67. _ + __ 2 2 x + 6x + 8
x  3x  28
x + 8x  20
x + 7x  18
3x + 12x
x  2x  24
Solve each equation or inequality. Round to the nearest tenthousandth. (Lesson 86) 70. 8x = 30
72. 3x + 2 = 41
71. 5x = 64
Evaluate each expression. (Lesson 76) 73. 16
_4
1 _
74. 27 3
4
75. 25
5 _ 2
Skills Review Solve for x. (Concepts and Skills Bank 1) x+2 x2 76. _ = _ 18
9
x+5 7 77. _ = _ x1
4
15 5 78. _ =_ x+8
2x + 20
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Law of Sines Then
Now
Why?
You found side lengths and angle measures of right triangles.
1
Find the area of a triangle using two sides and an included angle.
2
Use the Law of Sines to solve triangles.
Mars has hundreds of thousands of craters. These craters are named after famous scientists, science fiction authors, and towns on Earth. The craters named Wahoo, Wabash, and Naukan are shown in the figure. You can use trigonometry to find the distance between Wahoo and Naukan.
(Lesson 131)
NewVocabulary Law of Sines solving a triangle ambiguous case
Tennessee Curriculum Standards CLE 3103.4.4 Know and use the Law of Sines to find missing sides and angles of a triangle, including the ambiguous case. CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
1
8BIPP
1.2 km
?
102°
θ 23°
8BCBTI /BVLBO
Find the Area of a Triangle In the triangle at h the right, sin A = _ , or h = c sin A.
#
c
1 Area = _ bh
c
Formula for area of a triangle
2 _ Area = 1 b(c sin A) 2 _ Area = 1 bc sin A 2
Replace h with c sin A.
a
h b
"
$
Simplify.
You can use this formula or two other formulas to find the area of a triangle if you know the lengths of two sides and the measure of the included angle.
KeyConcept Area of a Triangle #
The area of a triangle is one half the product of the lengths of two sides and the sine of their included angle.
Words
Symbols
1 1 1 Area = _ bc sin A = _ ac sin B = _ ab sin C 2
2
c
a
b
"
$
2
Example 1 Find the Area of a Triangle Find the area of ABC to the nearest tenth.
#
In ABC, a = 8, b = 9, and C = 104°. 1 Area = _ ab sin C
2 1 =_ (8)(9) sin 104° 2
≈ 34.9 cm
2
MENTAL CHECK
Based on the known measures, use the third area formula.
8 cm
Substitution Simplify.
104°
"
9 cm
Round the sin 104° to sin 90° because the sin of 90° is 1.
_1 (8)(9)sin 90° = _1 (8)(9)(1) = 36 2
2
This is close to the answer of 34.9 square centimeters.
GuidedPractice 1. Find the area of ABC to the nearest tenth if A = 31°, b = 18 meters, and c = 22 meters.
832  Lesson 134
$
2
Use the Law of Sines to Solve Triangles You can use the area formulas to derive
the Law of Sines, which shows the relationships between side lengths of a triangle and the sines of the angles opposite them.
_1 bc sin A = _1 ac sin B = _1 ab sin C 2
2
Set the area formulas equal to each other.
2
bc sin A = ac sin B = ab sin C
Multiply each expression by 2.
ac sin B ab sin C bc sin A _ = _ =_
Divide each expression by abc.
abc sin A _ = a
abc sin B _ b
abc sin C _ = c
Simplify.
KeyConcept Law of Sines Math HistoryLink Pauline Sperry (1885–1967) Pauline Sperry was born in Peabody, Massachusetts. During the 1920s, she wrote two textbooks, Short Course in Spherical Trigonometry and Plane Trigonometry. In 1923, she became the first woman to be promoted to assistant professor in the mathematics department at Berkeley.
In ABC, if sides with lengths a, b, and c are opposite angles with measures A, B, and C, respectively, then the following is true.
# c
sin A sin B sin C _ =_ =_ b a c
a
b
"
$
You can use the Law of Sines to solve a triangle if you know either one of the following. • the measures of two angles and any side (angleangleside AAS or anglesideangle ASA cases)
# c
$
• the measures of two sides and the angle opposite one of the sides (sidesideangle SSA case)
c b
$
Using given measures to find all unknown side lengths and angle measures of a triangle is called solving a triangle.
StudyTip Alternative Representations The Law of Sines may also be written as a b c _ =_ =_ . sin A
sin B
sin C
So, the expressions below could also be used to solve the triangle in Example 2. 3 a • _ =_ sin 55° sin 80° 3 b • _ =_ sin 45° sin 80°
Example 2 Solve a Triangle Given Two Angles and a Side Solve ABC. Round to the nearest tenth if necessary.
"
Step 1 Find the measure of the third angle. m∠A = 180  (80 + 45) or 55°
80°
Step 2 Use the Law of Sines to find side lengths a and b. Write an equation to find each variable. sin A sin C _ =_
a c sin 55° sin 80° _ =_ 3 a 3 sin 55° a=_ sin 80°
a ≈ 2.5
Law of Sines
3
b
$
45° a
#
sin C sin B _ =_
c b sin 45° sin 80° _ Substitution =_ 3 b 3 sin 45° Solve for each variable. b=_ sin 80° Use a calculator.
b ≈ 2.2
So, A = 55°, a ≈ 2.5, and b ≈ 2.2.
GuidedPractice 2. Solve NPQ if P = 42°, Q = 65°, and n = 5.
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StudyTip Two Solutions A situation in which two solutions for a triangle exist is called the ambiguous case.
If you are given the measures of two angles and a side, exactly one triangle is possible. However, if you are given the measures of two sides and the angle opposite one of them, zero, one, or two triangles may be possible. This is known as the ambiguous case. So, when solving a triangle using the SSA case, zero, one, or two solutions are possible.
KeyConcept Possible Triangles in SSA Case Consider a triangle in which a, b, and m∠A are given. ∠A is Acute.
∠A is Right or Obtuse.
StudyTip A is Acute In the figures at the right, the altitude h is compared to a because h is the minimum distance from −− C to AB when A is acute.
a
a
b
b
a=h
h
"
"
a
opp hyp h _ sin A = b
sin A = _
b
b
a
h
"
a=h one solution b
a
h
a≤b no solution
a
"
"
h
a
b
"
a>b one solution
a≥b one solution
h Since sin A = _ , you can use h = b sin A to find h in the acute triangles. b
Example 3 Solve a Triangle Given Two Sides and an Angle Determine whether each triangle has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. a. In RST, R = 105°, r = 9, and s = 6. Because ∠R is obtuse and 9 > 6, you know that one solution exists. Step 1 Use the Law of Sines to find m∠S. sin S sin 105° _ =_
Law of Sines
6 sin 105° sin S = _
Multiply each side by 6.
sin S ≈ 0.6440 S ≈ 40°
Use a calculator. Use the sin1 function.
6
9
9
Step 2 Find m∠T. m∠T ≈ 180  (105 + 40) or 35° Step 3 Use the Law of Sines to find t. sin 35° sin 105° _ ≈_ t
9 9 sin 35° _ t≈ sin 105°
t ≈ 5.3
Law of Sines Solve for t. Use a calculator.
So, S ≈ 40°, T ≈ 35°, and t ≈ 5.3.
834  Lesson 134  Law of Sines
4 9 t 105°
3
6
5
b. In ABC, A = 54°, a = 6, and b = 8.
$
Since ∠A is acute and 6 < 8, find h and compare it to a. b sin A = 8 sin 54°
b = 8 and A = 54°
≈ 6.5
6
8
h
54°
"
Use a calculator.
Since 6 ≤ 6.5 or a ≤ h, there is no solution. c. In ABC, A = 35°, a = 17, and b = 20.
$ 20
Since ∠A is acute and 17 < 20, find h and compare it to a. b sin A = 20 sin 35° ≈ 11.5
b = 20 and A = 35°
17
h
35°
"
Use a calculator.
Since 11.5 < 17 < 20 or h < a < b, there are two solutions. So, there are two triangles to be solved ∠B is acute.
Case 1
∠B is obtuse.
Case 2
$ 20
$ 20
17
35°
StudyTip Reference Angle In the triangle in Case 2, you are using the reference angle 42° to find the other value of B.
"
Step 1
#
sin B ≈ 0.6748 B ≈ 42° Step 2
"
Step 1
Find m∠B.
sin 35° sin B _ =_ 17 20 20 sin 35° sin B = _ 17
17
35°
Law of Sines Solve for sin B. Use a calculator. Find sin1 0.6748.
Find m∠B.
The sine function also has a positive value in Quadrant II. So, find an obtuse angle B for which sin B ≈ 0.6748. m∠B ≈ 180°  42° or 138° Step 2
Find m∠C.
#
Find m∠C.
m∠C ≈ 180  (35 + 42) or 103°
m∠C ≈ 180  (35 + 138) or 7°
Step 3
Step 3
Find c.
sin 103° 35° _ _ = sin c 17 sin 103° _ c = 17 sin 35°
c ≈ 28.9
Law of Sines Solve for c. Simplify.
Find c.
sin 7° 35° _ _ ≈ sin c 17 sin 7° _ c ≈ 17 sin 35°
c ≈ 3.6
Law of Sines Solve for c. Simplify.
So, one solution is B ≈ 42°, C ≈ 103°, and c ≈ 28.9, and another solution is B ≈ 138°, C ≈ 7°, and c ≈ 3.6.
GuidedPractice Determine whether each triangle has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 3A. In RST, R = 95°, r = 10, and s = 12. 3B. In MNP, N = 32°, n = 7, and p = 4. 3C. In ABC, A = 47°, a = 15, and b = 18.
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RealWorld Example 4 Use the Law of Sines to Solve a Problem BASEBALL A baseball is hit between second and third bases and is caught at point B, as shown in the figure. How far away from second base was the ball caught? sin 72° sin 43° _ =_ 90
x
x sin 72° = 90 sin 43°
High school and college baseball fields share the same infield dimensions as professional baseball fields. The outfield dimensions vary greatly.
OE CBTF 65°
Law of Sines Cross products 43°
90 sin 43° x=_
Solve for x.
x ≈ 64.5
Use a calculator.
sin 72°
RealWorldLink
#
90 ft SECBTF
So, the distance is about 64.5 feet.
GuidedPractice 4. How far away from third base was the ball caught?
Source: Baseball Digest Magazine
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R20.
Find the area of ABC to the nearest tenth, if necessary. 1.
" 86°
2.
7 mm
#
#
4 yd
30°
"
3 yd
8 mm
$ $
3 A = 40°, b = 11 cm, c = 6 cm Example 2
4. B = 103°, a = 20 in., c = 18 in.
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 5.
% 12
6.
d
a
97°
39° f
34°
'
#
&
$
c 50°
9
"
7. Solve FGH if G = 80°, H = 40°, and g = 14. Example 3
Determine whether each ABC has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 8. A = 95°, a = 19, b = 12 8BIPP
9. A = 60°, a = 15, b = 24 10. A = 34°, a = 8, b = 13
1.2 km
102°
11. A = 30°, a = 3, b = 6
? θ 23°
8BCBTI
Example 4
12. SPACE Refer to the beginning of the lesson. Find the distance between the Wahoo Crater and the Naukan Crater on Mars.
836  Lesson 134  Law of Sines
/BVLBO
Practice and Problem Solving Example 1
Extra Practice begins on page 947
Find the area of ABC to the nearest tenth. 13. "
14.
"
# 6 km
52° 20 ft
16 ft
5 km 45°
#
$
$
15. $
16.
8m 113°
30° 10 m
#
$
14 cm 36° 93°
#
18 cm
51°
" "
Example 2
17. C = 25°, a = 4 ft, b = 7 ft
18. A = 138°, b = 10 in., c = 20 in.
19. B = 92°, a = 14.5 m, c = 9 m
20. C = 116°, a = 2.7 cm, b = 4.6 cm
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 21.
#
$
8
22. 4
106° c
44°
r
47° t
b
5
53° 13
3 "
23.
24. #
. n
70°
$ 112°
5 c
36°

a
m
24 30°
/ "
25 Solve HJK if H = 53°, J = 20°, and h = 31. 26. Solve NPQ if P = 109°, Q = 57°, and n = 22. 27. Solve ABC if A = 50°, a = 2.5, and C = 67°. 28. Solve ABC if B = 18°, C = 142°, and b = 20.
Example 3
Determine whether each ABC has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 29. A = 100°, a = 7, b = 3
30. A = 75°, a = 14, b = 11
31. A = 38°, a = 21, b = 18
32. A = 52°, a = 9, b = 20
33. A = 42°, a = 5, b = 6
34. A = 44°, a = 14, b = 19
35. A = 131°, a = 15, b = 32
36. A = 30°, a = 17, b = 34 connectED.mcgrawhill.com
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Example 4
GEOGRAPHY In Hawaii, the distance from Hilo to Kailua is 57 miles, and the distance from Hilo to Captain Cook is 55 miles.
Kailua
37. What is the measure of the angle formed at Hilo?
Captain Cook
Hilo
57 mi 80°
55 mi
38. What is the distance between Kailua and Captain Cook?
B
39 TORNADOES Tornado sirens A, B, and C form a triangular region in one area of a city. Sirens A and B are 8 miles apart. The angle formed at siren A is 112°, and the angle formed at siren B is 40°. How far apart are sirens B and C? 40. MYSTERIES The Bermuda Triangle is a region of the Atlantic Ocean between Bermuda, Miami, Florida, and San Juan, Puerto Rico. It is an area where ships and airplanes have been rumored to mysteriously disappear.
Bermuda
Miami
965 mi
53°
a. What is the distance between Miami and Bermuda?
1038 mi
b. What is the approximate area of the Bermuda Triangle?
San Juan
41. BICYCLING One side of a triangular cycling path is 4 miles long. The angle opposite this side is 64°. Another angle formed by the triangular path measures 66°. a. Sketch a drawing of the situation. Label the missing sides a and b. b. Write equations that could be used to find the lengths of the missing sides. c. What is the perimeter of the path?
h
42. ROCK CLIMBING Savannah S and Leon L are standing 8 feet apart in front of a rock climbing wall, as shown at the right. What is the height of the wall? Round to the nearest tenth.
H.O.T. Problems C
45° S
60°
8 ft L
Use HigherOrder Thinking Skills
43. ERROR ANALYSIS In RST, R = 56°, r = 24, and t = 12. Cameron and Gabriela are using the Law of Sines to find T. Is either of them correct? Explain your reasoning.
Gabriela
Cameron sin T _ sin 56° _ = 12
Since r > t, there is no solution.
24
sin T ≈ 0.4145 T ≈ 24.5°
44. OPEN ENDED Create an application problem involving right triangles and the Law of Sines. Then solve your problem, drawing diagrams if necessary.
#
45. CHALLENGE Using the figure at the right, derive 1 the formula Area = _ bc sin A.
c
a
h
2
46. REASONING Find the side lengths of two different triangles ABC that can be formed if A = 55° and C = 20°.
b
"
$ #
47. WRITING IN MATH Use the Law of Sines to explain why a and b do not have unique values in the figure shown. 48. OPEN ENDED Given that E = 62° and d = 38, find a value for e such that no triangle DEF can exist. Explain your reasoning.
838  Lesson 134  Law of Sines
a 50°
"
b
$
SPI 3103.3.6, SPI 3102.5.2, SPI 3103.3.2
Standardized Test Practice 51. One zero of f(x) = x3  7x2  6x + 72 is 4. What is the factored form of the expression x3  7x2  6x + 72?
49. SHORT RESPONSE Given the graphs of f(x) and g(x), what is the value of f(g(4))? f (x )
8 6 4 2
y
−8−6−4−20
2 4 6 8x
−4 −6 −8
g (x )
F G H J
52. SAT/ACT Three people are splitting $48,000 using the ratio 5 : 4 : 3. What is the amount of the greatest share?
50. STATISTICS If the average of seven consecutive odd integers is n, what is the median of these seven integers?
A $12,000 B $16,000 C $20,000
C n D n2
A 0 B 7
(x  6)(x + 3)(x + 4) (x  6)(x + 3)(x  4) (x + 6)(x + 3)(x  4) (x + 12)(x  1)(x  4)
D $24,000 E $30,000
Spiral Review Find the exact value of each trigonometric function. (Lesson 133) 3 54. cos _ π
53. sin 210°
55. cot 60°
4
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. (Lesson 132) 2 58. _ π
57. 32º
56. 125º
3
59. CLOCKS Jun’s grandfather clock is broken. When she sets the pendulum in motion by holding it against the side of the clock and letting it go, it swings 24 centimeters to the other side, then 18 centimeters back, then 13.5 centimeters, and so on. What is the total distance that the pendulum swings before it stops? (Lesson 115) Find the sum of each infinite series, if it exists. (Lesson 114) 60. 64 + 48 + 36 + …
∞
61. 27 + 36 + 48 + …
62. ∑ 0.5(1.1)n n=1
63. ASTRONOMY At its closest point, Earth is 91.8 million miles from the center of the Sun. At its farthest point, Earth is 94.9 million miles from the center of the Sun. Write an equation for the orbit of Earth, assuming that the center of the orbit is the origin and the Sun lies on the xaxis. (Lesson 104) Simplify. (Lesson 74) 64.
(x  4)2 √
65.
(y + 2)4 √
66.
3
(a  b)6 √
Skills Review 3 Evaluate each expression if w = 6, x = 4, y = 1.5, and z = _ . (Lesson 11) 4
67.
w2
+
y2
 6xz
68.
x2
+
z2
+ 5wy
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Geometry Lab
Regular Polygons You can use central angles of circles to investigate characteristics of regular polygons inscribed in a circle. Recall that a regular polygon is inscribed in a circle if each of its vertices lies on the circle.
Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
Activity
Collect the Data
Step 1 Use a compass to draw a circle with a radius of one inch. Step 2 Inscribe an equilateral triangle inside the circle. To do this, use a protractor 360º to measure three angles of 120º at the center of the circle, since _ = 120º. 3
Then connect the points where the sides of the angles intersect the circle using a straightedge. Step 3 The apothem of a regular polygon is a segment that is drawn from the center of the polygon perpendicular to a side of the polygon. Use the cosine of angle θ to find the length of an apothem, labeled a in the diagram.
Model and Analyze 1. Make a table like the one shown below and record the length of the apothem of the equilateral triangle. Inscribe each regular polygon named in the table in a circle with radius one inch. Copy and complete the table. Number of Sides, n
θ
3
60
7
4
45
8
a
Number of Sides, n
5
9
6
10
θ
a 1 in. 120°
a θ
2. What do you notice about the measure of θ as the number of sides of the inscribed polygon increases? 3. What do you notice about the value of a? 4. MAKE A CONJECTURE Suppose you inscribe a 30sided regular polygon inside a circle. Find the measure of angle θ. 5. Write a formula that gives the measure of angle θ for a polygon with n sides. 6. Write a formula that gives the length of the apothem of a regular polygon inscribed in a circle with radius one inch. 7. How would the formula you wrote in Exercise 5 change if the apothem of the circle was not one inch?
840  Extend 134  Geometry Lab: Regular Polygons
Law of Cosines Then
Now
Why?
You solved triangles by using the Law of Sines. (Lesson 134)
1
Use the Law of Cosines to solve triangles.
2
Choose methods to solve triangles.
A submersible is an underwater vessel used for exploring the depths of the ocean. You can use trigonometry to find the distance from a ship used to lower a submersible into the ocean and a shipwreck spotted by the submersible on the ocean floor.
NewVocabulary Law of Cosines
Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
1
Use Law of Cosines to Solve Triangles You cannot use the Law of Sines to solve
a triangle like the one shown above. You can use the Law of Cosines if: • the measures of two sides and the included angle are known (sideangleside case). • the measures of three sides are known (sidesideside case).
KeyConcept Law of Cosines In ABC, if sides with lengths a, b, and c are opposite angles with measures A, B, and C, respectively, then the following are true. a 2 = b 2 + c 2  2bc cos A b 2 = a 2 + c 2  2ac cos B c 2 = a 2 + b 2  2ab cos C
# c
a
b
"
$
Example 1 Solve a Triangle Given Two Sides and the Included Angle Solve ABC.
A
Step 1 Use the Law of Cosines to find the missing side length. b2 = a2 + c2  2ac cos B b2 = 72 + 52  2(7)(5) cos 36° b2 ≈ 17.4 b ≈ 4.2
b 5
C
Law of Cosines a = 7, c = 5, B = 36°
7
Use a calculator to simplify. Take the positive square root of each side.
36°
B
Step 2 Use the Law of Sines to find a missing angle measure. sin A sin 36° _ ≈_ 7
4.2 7 sin 36° sin A ≈ _ 4.2
A ≈ 78°
sin A sin B _ =_ a
b
Multiply each side by 7. Use the sin1 function.
Step 3 Find the measure of the other angle. m∠C ≈ 180°  (36° + 78°) or 66° So, b ≈ 4.2, A ≈ 78°, and C ≈ 66°.
GuidedPractice 1. Solve FGH if G = 82°, f = 6, and h = 4.
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When you are only given the three side lengths of a triangle, you can solve it by using the Law of Cosines. The first step is to find the measure of the largest angle. This is done to ensure the other two angles are acute when using the Law of Sines.
Example 2 Solve a Triangle Given Three Sides Solve ABC.
"
10
#
9 16
$
Step 1 Use the Law of Cosines to find the measure of the largest angle, ∠A. a2 = b2 + c2  2bc cos A 162

92

162
92
=
102
= 2(9)(10) cos A
+
102
Law of Cosines
 2(9)(10) cos A
a = 16, b = 9, and c = 10 Subtract 92 and 102 from each side.
162  92  102 __ = cos A
Divide each side by 2(9)(10).
2(9)(10)
0.4167 ≈ cos A
Use the cos1 function.
115° ≈ A
StudyTip Alternative Method After finding m∠A in Step 1, the Law of Cosines could be used again to find the measure of a second angle.
Use a calculator to simplify.
Step 2 Use the Law of Sines to find the measure of ∠B. sin B sin 115° _ ≈_ 9
16 9 sin 115° sin B ≈ _ 16
sin B ≈ 0.5098
sin B sin A _ =_ b
a
Multiply each side by 9. Use a calculator. Use the sin1 function.
B ≈ 31° Step 3 Find the measure of ∠C.
m∠C ≈ 180°  (115° + 31°) or about 34° So, A ≈ 115°, B ≈ 31°, and C ≈ 34°.
GuidedPractice 2. Solve ABC if a = 5, b = 11, and c = 8.
ReviewVocabulary oblique a triangle that has no right angle
2
Choose a Method to Solve Triangles You can use the Law of Sines and the Law of
Cosines to solve problems involving oblique triangles. You need to know the measure of at least one side and any two other parts. If the triangle has a solution, you must decide whether to use the Law of Sines or the Law of Cosines to begin solving it.
ConceptSummary Solving Oblique Triangles Given G
Begin by Using
two angles and any sides
Law of Sines
two sides and an angle opposite one of them
Law of Sines
two sides and their included angle
Law of Cosines
three sides
Law of Cosines
842  Lesson 135  Law of Cosines
RealWorld Example 3 Use the Law of Cosines SCUBA DIVING A scuba diver looks up 20° and sees a turtle 9 feet away. She looks down 40° and sees a blue parrotfish 12 feet away. How far apart are the turtle and the blue parrotfish? Understand You know the angles formed when the scuba diver looks up and when she looks down. You also know how far away the turtle and the blue parrotfish are from the scuba diver.
UVSUMF # TDVCB 9 ft EJWFS 20° " 40°
Plan Use the information to draw and label a diagram. Since two sides and the included angle of a triangle are given, you can use the Law of Cosines to solve the problem. Solve a2 = b2 + c2  2bc cos A a2 = 122 + 92  2(12)(9) cos 60 a2 = 117 a ≈ 10.8
RealWorldLink The record for the deepest seawater scuba dive was 1044 feet, made by a scuba diver in the Red Sea.
B
12 ft CMVF QBSSPUGJTI
$
Law of Cosines b = 12, c = 9, and A = 60 Use a calculator. Find the positive value of a.
So, the turtle and the blue parrotfish are about 10.8 feet apart.
Source: Guinness Book of World Records
Check Using the Law of Sines, you can find that B ≈ 74° and C ≈ 46°. Since C < A < B and c < a < b, the solution is reasonable.
GuidedPractice 3. MARATHONS Amelia ran 6 miles in one direction. She then turned 79° and ran 7 miles. At the end of the run, how far was Amelia from her starting point?
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. #
1. 4
"
2.
92°
3
"
10
#
14 20
b
$ $
3. a = 5, b = 8, c = 12 Example 3
4. B = 110°, a = 6, c = 3
Determine whether each triangle should be solved by beginning with the Law of Sines or the Law of Cosines. Then solve the triangle. 5. #
6. 12
c
4
5
96°
$ b
107°
"
#
8
$
"
7 In RST, R = 35°, s = 16, and t = 9. 8. FOOTBALL In a football game, the quarterback is 20 yards from Receiver A. He turns 40° to see Receiver B, who is 16 yards away. How far apart are the two receivers? connectED.mcgrawhill.com
843
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. $
9.
70°
3
2
14
$
12
" 7
#
11.
92°
b
c
#
#
10.
"
12.
$
" 10
9 13
8 14
$
#
"
Example 3
13. A = 116°, b = 5, c = 3
14. C = 80°, a = 9, b = 2
15. f = 10, g = 11, h = 4
16. w = 20, x = 13, y = 12
Determine whether each triangle should be solved by beginning with the Law of Sines or the Law of Cosines. Then solve the triangle. "
17. 14
13 a
$ #
19. 15
4
18.
11
16
20
106°
5 s
50°
# $
3
20. .
p
31
22
80°
m
1
"
21. In ABC, C = 84°, c = 7, and a = 2.
/
47°
22. In HJK, h = 18, j = 10, and k = 23.
23 EXPLORATION Find the distance between the ship and the shipwreck shown in the diagram. Round to the nearest tenth. 24. GEOMETRY A parallelogram has side lengths 8 centimeters and 12 centimeters. One angle between them measures 42°. To the nearest tenth, what is the length of the shorter diagonal?
B
520 m 70°
?
338 m
25. RACING A triangular crosscountry course has side lengths 1.8 kilometers, 2 kilometers, and 1.2 kilometers. What are the angles formed between each pair of sides? 26. SURVEYING A triangular plot of farm land measures 0.9 by 0.5 by 1.25 miles. a. If the plot of land is fenced on the border, what will be the angles at which the fences of the three sides meet? Round to the nearest degree. b. What is the area of the plot of land? 27. LAND Some land is in the shape of a triangle. The distances between each vertex of the triangle are 140 yd, 210 yd and 300 yd, respectively. Use the Law of Cosines to find the area of the land to the nearest square yard.
844  Lesson 135  Law of Cosines
28. RIDES Two bumper cars at an amusement park ride collide as shown below.
d 7 ft 5.5 ft
118°
a. How far apart d were the two cars before they collided? b. Before the collision, a third car was 10 feet from car 1 and 13 feet from car 2. Describe the angles formed by cars 1, 2, and 3 before the collision.
C
29. PICNICS A triangular picnic area is 11 yards by 14 yards by 10 yards. a. Sketch and label a drawing to represent the picnic area. b. Describe how you could find the area of the picnic area. c. What is the area? Round to the nearest tenth. 30. WATERSPORTS A person on a personal watercraft makes a trip from point A to point B to point C traveling 28 miles per hour. She then returns from point C back to her starting point traveling 35 miles per hour. How many minutes did the entire trip take? Round to the nearest tenth.
# 0.25 mi
130°
0.15 mi
$
"
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 31 #
32. 2
28 25°
12.4
c
H.O.T. Problems
20.8
33. '
(
q
36.2
15.2
21.6
4
104°
"
3
8.1
)
$
Use HigherOrder Thinking Skills #
34. CHALLENGE Use the figure and the Pythagorean Theorem to derive the Law of Cosines. Use the hints below. • First, use the Pythagorean Theorem for DBC. • In ADB, c2 = x2 + h2.
c
"
h x
%
a
bx
$
b
x • cos A = _ c
35. REASONING Three sides of a triangle measure 10.6 centimeters, 8 centimeters, and 14.5 centimeters. Explain how to find the measure of the largest angle. Then find the measure of the angle to the nearest degree. 36. OPEN ENDED Create an application problem involving right triangles and the Law of Cosines. Then solve your problem, drawing diagrams if necessary. 37.
E WRITING IN MATH Compare the circumstances in which you can use the Law of Sines and the Law of Cosines to solve a triangle. connectED.mcgrawhill.com
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SPI 3103.3.13, SPI 3108.4.3
Standardized Test Practice 38. SAT/ACT If c and d are different positive integers and 4c + d = 26, what is the sum of all possible values of c? A 6 B 10 C 15
40. GEOMETRY Find the perimeter of the figure.
12
D 21 E 28
60°
A 24 39. If
6y
12
B 30
C 36
D 48
= 21, what is y?
F log 12  log 6 log 21 G _
41. SHORT RESPONSE Solve the equation below for x.
log 6 H _
5 23 1 _ +_ =_
log 21
x1
6 J log _ 21
( )
log 6
8
6x
Spiral Review Find the area of ABC to the nearest tenth. (Lesson 134) 42.
" 81°
11 cm
#
43. #
12 cm
5 yd
30° 6 yd
"
"
44. 8 ft
12 ft
$
$
$
47°
#
The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ. (Lesson 133) 45. (8, 5)
46. (4, 2)
47. (6, 9)
48. EDUCATION The Millersburg school board is negotiating a pay raise with the teachers’ union. Three of the administrators have salaries of $90,000 each. However, a majority of the teachers have salaries of about $45,000 per year. (Lesson 122) a. You are a member of the school board and would like to show that the current salaries are reasonable. Would you quote the mean, median, or mode as the “average” salary to justify your claim? Explain. b. You are the head of the teachers’ union and maintain that a pay raise is in order. Which of the mean, median, or mode would you quote to justify your claim? Explain your reasoning. 49. BUSINESS During the month of June, MediaWorld had revenue of $2700 from sales of a certain DVD box set. During the July Blowout Sale, the set was on sale for $10 off. Revenue from the set was $3750 in July with 30 more sets sold than were sold in June. Find the price of the DVD set for June and the price for July. (Lesson 107) Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. (Lesson 106) 50. x2 + y2  8x  6y + 5 = 0
51. 3x2  2y2 + 32y  134 = 0
52. y2 + 18y  2x = 84
Skills Review Sketch each angle. Then find its reference angle. (Lesson 133) 53. 245º
846  Lesson 135  Law of Cosines
54. 15º
5 55. _ π 4
MidChapter Quiz
Tennessee Curriculum Standards
Lessons 131 through 135
SPI 3108.4.8, SPI 3103.4.1
Solve XYZ by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. (Lesson 131)
14. GARDEN Lana has a garden in the shape of a triangle as pictured below. She wants to fill the garden with top soil. What is the area of the triangle? (Lesson 134)
: z
9
8m
x
44°
y
10 m
;
1. Y = 65°, x = 16
2. X = 25°, x = 8
3. Find the values of the six trigonometric functions for angle θ.
15. A = 38°, a = 18, c = 25
12
16. A = 65°, a = 5, b = 7
θ
17. A = 115°, a = 12, b = 8
9
4. Draw an angle measuring 80° in standard position. (Lesson 132)
5. 215°
6. 350°
8π 7. _
9π 8. _ 2
9. MULTIPLE CHOICE What is the length of the arc below rounded to the nearest tenth? (Lesson 132)
B 17.1 cm
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 135) 18.
Rewrite each degree measure in radians and each radian measure in degrees. (Lesson 132)
A 4.2 cm
Determine whether each triangle has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures the nearest degree. (Lesson 134)
(Lesson 131)
5
#
10. tan π
4
18
c
24.2
105°
#
12
$
20. Eric and Zach are camping. Erik leaves Zach at the campsite and walks 4.5 miles. He then turns at a 120° angle and walks another 2.5 miles. If Eric were to walk directly back to Zach, how far would he walk? (Lesson 135) &SJD E 2.5 mi
15 cm
3π 11. cos _
$
"
8π 7
Find the exact value of each trigonometric function. (Lesson 133)
19. "
18.6
16.4
C 53.9 cm D 2638.9 cm
60°
120°
4.5 mi
;BDI
21. MULTIPLE CHOICE Suppose θ is an angle in standard position with cos θ > 0. In which quadrant(s) does the terminal side of θ lie? (Lesson 132) F I
The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ.
G II
(Lesson 133)
H III
12. (0, 5)
13. (6, 8)
J I and IV connectED.mcgrawhill.com
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Circular Functions Then
Now
Why?
You evaluated trigonometric functions using reference angles.
1
Find values of trigonometric functions based on the unit circle.
The pedals on a bicycle rotate as the bike is being ridden. The height of a pedal is a function of time, as shown in the figure at the right.
2
Use the properties of periodic functions to evaluate trigonometric functions.
Notice that the pedal makes one complete rotation every two seconds.
(Lesson 133)
NewVocabulary unit circle circular function periodic function cycle period
1
0.5s
18 in.
✔ 3103.1.7 Use the unit circle to determine the exact value of trigonometric functions for commonly used angles (0°, 30°, 45°, 60°…). CLE 3103.4.1 Understand the trigonometric functions and their relationship to the unit circle. ✔ 3103.4.3 Extend the trigonometric functions to periodic functions on the real line by defining them as functions on the unit circle. SPI 3103.4.1 Exhibit knowledge of unit circle trigonometry. Also addresses CLE 3103.4.3 and CLE 3103.4.5.
1.0s
4 in.
(0, 1)
a radius of 1 unit centered at the origin on the coordinate plane. You can use a point P on the unit circle to generalize sine and cosine functions. y y sin θ = _ = _ or y
x x cos θ = _ =_ or x r
1
(1, 0)
1 (x, y )
1
y
θ x
0
x (1, 0)
(0, 1)
KeyConcept Functions on a Unit Circle If the terminal side of an angle θ in standard position intersects the unit circle at P(x, y), then cos θ = x and sin θ = y.
Words
Model 1 (cos θ, sin θ)
(1, 0)
Symbols
P(x, y) = P(cos θ, sin θ)
Example
If θ = 120°, P(x, y) = P(cos 120°, sin 120°).
(0, 1)
y
θ
(1, 0) x
0
(0, 1)
Both cos θ = x and sin θ = y are functions of θ. Because they are defined using a unit circle, they are called circular functions.
Example 1 Find Sine and Cosine Given a Point on the Unit Circle The terminal side of angle θ in standard position intersects √ 3 1 _ the unit circle at P(_ , ). Find cos θ and sin θ.
(2 2 )
2
(0, 1)
1 cos θ = _ 2
y
1
( 12 , √23 )
2
√ 3 1 _ P _ , = P(cos θ, sin θ)
θ (1, 0)
√ 3 2
sin θ = _
0
(0, 1)
GuidedPractice 1. The terminal side of angle θ in standard position 3 4 intersects the unit circle at P _ , _ . Find cos θ and sin θ.
(5
848  Lesson 136
y
1
So, the values of sin θ and cos θ are the ycoordinate and xcoordinate, respectively, of the point where the terminal side of θ intersects the unit circle. Tennessee Curriculum Standards
1.5s
11 in.
Circular Functions A unit circle is a circle with
r
2.0s
0s
5
)
(1, 0) x
StudyTip Cycles A cycle can begin at any point on the graph of a periodic function. In Example 2, if the beginning of the
2
Periodic Functions A periodic function has yvalues that repeat at regular
intervals. One complete pattern is a cycle, and the horizontal length of one cycle is a period. y
π cycle is at _ , then the 2
3π . pattern repeats at _ 2 3π π _ or π. The period is _ 2 2
θ 0° 180° 360° 540° 720°
1 cycle
1 0
180° 360° 540° 720°
θ
1 period: 360°
y 1 1 1 1 1
The cycle repeats every 360°.
Example 2 Identify the Period y
Determine the period of the function. 1
The pattern repeats at π, 2π, and so on. So, the period is π.
0 1
π
π 2
3π 2
2π
θ
GuidedPractice 2. Graph a function with a period of 4.
T rotations of wheels, pedals, carousels, and objects in space are all periodic. The
RealWorld Example 3 Use Trigonometric Functions C CYCLING Refer to the beginning of the lesson. The height of a bicycle pedal varies periodically as a function of time, as shown in the figure. p a. Make a table showing the height of a bicycle pedal at 0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 seconds.
Most competitive cyclists pedal at rates of more than 200 rotations per minute. Most other people pedal at between 90 and 120 rotations per minute. Source: SpringerLink
At 0 seconds, the pedal is 18 inches high. At 0.5 second, the pedal is 11 inches high. At 1.0 second, the pedal is 4 inches high, and so on. b. Identify the period of the function. The period is the time it takes to complete one rotation. So, the period is 2 seconds.
The maximum height of the pedal is 18 inches, and the minimum height is 4 inches. Because the period of the function is 2 seconds, the pattern of the graph repeats in intervals of 2 seconds.
.5
Height (in.) 18 11 4 11 18 11 4
c. Graph the function. Let the horizontal axis represent the time t and the vertical axis represent the height h in inches that the pedal is from the ground. Height of Pedal (in.)
RealWorldLink
Time (s) 0 0.5 1.0 1.5 2.0 2.5 3.0
GuidedPractice
h 20 10 0
0.5
1
1.5
2
2.5
3 θ
Time (s)
3. CYCLING Another cyclist pedals the same bike at a rate of 1 revolution per second. A. Make a table showing the height of a bicycle pedal at times 0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 seconds. B. Identify the period and graph the function.
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StudyTip Sine and Cosine To help you remember that for (x, y) on a unit circle, x = cos θ and y = sin θ, notice that alphabetically x comes before y and cosine comes before sine.
The exact values of cos θ and sin θ for special angles are shown on the unit circle at the right. The cosine values are the xcoordinates of the points on the unit circle, and the sine values are the ycoordinates. You can use this information to graph the sine and cosine functions. Let the horizontal axis represent the values of θ and the vertical axis represent the values of sin θ or cos θ.
Z
((
(2 2) 3 1 (cos 150°, sin 150°) = (_, _ 2 2) √ 2 √ 2 (cos 45°, sin 45°) = _, _
√2
√2 , 2 2 √3 , 1 2 2
)
)
(0, 1)
)
√3
2
5π 6
π 2
( 12 ,
√3
2
π 3 π 90˚ 60˚ 4 π 45˚ 6
2π 3π 3 120˚ 4
135˚ 150˚ ** 180˚
) ( 22 , 22 ) ( 23 , 12 ) √
√
√
30˚ 0˚ 0 (1, 0) Y 360˚ 0 *** *7 330 ˚ √ 7π 210˚ √3 315˚ 11π  3,  1 225˚ ,1 6 2 2 300˚ 7π 6 2 2 5π 240˚ 270˚ 5π 4 √2 √2 √2 √2 4 4π  2 , 2 , 3 3π 3 2 2 (1, 0) π
( (
*
)
)
(
√  1,  3 2 2
The cycles of the sine and cosine functions repeat every 360°. So, they are periodic functions. The period of each function is 360° or 2π. Consider the points on the unit circle for θ = 45°, θ = 150°, and θ = 270°.
( 12 ,
)
1
(
√  3, 1 2 2
)
150° 0
1
√
(cos 270°, sin 270°) = (0, 1)
( ( 12 , 
2
(0, 1)
y
(
√3
2
)
( √22 , √22 ) x
45° 1
1 (0, 1)
StudyTip Radians The sine and cosine functions can also be graphed using radians as the units on the θaxis.
These points can also be shown on the graphs of the sine and cosine functions. y 1
sin 45° =
y
√2 2
1
1
sin 150° = 2
0
90°
1
180° 270° 360° θ
0 1
sin 270° = 1
cos 45° =
90°
√2 2
cos 270° = 0
180° 270° 360° θ √3 2
cos 150° =
Since the period of the sine and cosine functions is 360°, the values repeat every 360°. So, sin (x + 360°) = sin x, and cos (x + 360°) = cos x.
Example 4 Evaluate Trigonometric Functions Find the exact value of each function. 11π b. sin _
a. cos 480°
4
cos 480° = cos (120° + 360°)
3π 8π 11π sin _ = sin _ +_
= cos 120°
3π = sin _
1 = _
√ 2 = _ 2
2
(4
4
4
GuidedPractice 3π 4A. cos _
(
850  Lesson 136  Circular Functions
4
)
4B. sin 420°
4
)
) )
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R20.
The terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ.
(
15 _ 1. P _ , 8
Example 2
)
Determine the period of each function. y
3.
y
4.
1
1
0
2
4
6
8
x
0
1
Example 3
√ 2 2 √ 2 2
2. P _, _
( 17 17 )
π
2π
4π θ
3π
1
5. SWINGS The height of a swing varies periodically as the function of time. The swing goes forward and reaches its high point of 6 feet. It then goes backward and reaches 6 feet again. Its lowest point is 2 feet. The time it takes to swing from its high point to its low point is 1 second. a. How long does it take for the swing to go forward and back one time? b. Graph the height of the swing h as a function of time t.
Example 4
Find the exact value of each function. 13π 6. sin _
7. sin (–60°)
6
8. cos 540°
Practice and Problem Solving Example 1
Extra Practice begins on page 947.
The terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ. 6 8 9. P _ , _
( 10
10
10 24 10. P _ , _
(
)
26
(5
( 2 2)
√ 19 6 √ 12. P _, _
√ 3 1 11 P _, _
Example 2
26
5
)
)
Determine the period of each function. y
13.
0
2
4
6
8
x
y
15.
1
0
2 4 6 8 10 12 14 16
x
y
16. 1
1
0
y
14.
2 4 6 8 10 12 14 16 18 20 x
0
2
4
6
8
10
12 x
1
connectED.mcgrawhill.com
851
Determine the period of each function. y
17. 1
1
0
180°
360°
540°
720° θ
1
Example 3
y
18.
0
π
2π
4π θ
3π
1
19. WEATHER In a city, the average high temperature for each month is shown in the table. a. Sketch a graph of the function representing this situation. b. Describe the period of the function.
Average High Temperatures Month Jan Feb. Mar. Apr. May Jun.
Temperature (°F) 36 41 52 64 74 82
Month July Aug. Sept. Oct. Nov. Dec.
Temperature (°F) 85 84 78 66 52 41
Source: The Weather Channel
Example 4
Find the exact value of each function. 7π 20. sin _
21. cos (60°)
22. cos 450°
11π 23. sin _
24. sin (45°)
25. cos 570°
3
4
B
26. ENGINES In the engine at the right, the distance d from the piston to the center of the circle, called the crankshaft, is a function of the speed of the piston rod. Point R on the piston rod rotates 150 times per second. a. Identify the period of the function as a fraction of a second. b. The shortest distance d is 0.5 inch, and the longest distance is 3.5 inches. Sketch a graph of the function. Let the horizontal axis represent the time t. Let the vertical axis represent the distance d.
piston piston d
d 3
3
27 TORNADOES A tornado siren makes 2.5 rotations per minute and the beam of sound has a radius of 1 mile. Ms. Miller’s house is 1 mile from the siren. The distance of the sound beam from her house varies periodically as a function of time. a. Identify the period of the function in seconds. b. Sketch a graph of the function. Let the horizontal axis represent the time t from 0 seconds to 60 seconds. Let the vertical axis represent the distance d the sound beam is from Ms. Miller’s house at time t. 28. FERRIS WHEEL A Ferris wheel in China has a diameter of approximately 520 feet. The height of a compartment h is a function of time t. It takes about 30 seconds to make one complete revolution. Let the height at the center of the wheel represent the height at time 0. Sketch a graph of the function.
852  Lesson 136  Circular Functions
29.
MULTIPLE REPRESENTATIONS The terminal side of an angle in standard position intersects the unit circle at P, as shown in the figure.
1
y
1 120°
a. Geometric Copy the figure. Draw lines representing 30º, 60º, 150º, 210º, and 315º.
1x
0
1
b. Tabular Use a table of values to show the slope of each line to the nearest tenth.
1
c. Analytical What conclusions can you make about the relationship between the terminal side of the angle and the slope? Explain your reasoning. 30. POGO STICK A person is jumping up and down on a pogo stick at a constant rate. The difference between his highest and lowest points is 2 feet. He jumps 50 times per minute. a. Describe the independent variable and dependent variable of the periodic function that represents this situation. Then state the period of the function in seconds. b. Sketch a graph of the jumper’s change in height in relation to his starting point. Assume that his starting point is halfway between his highest and lowest points. Let the horizontal axis represent the time t in seconds. Let the vertical axis represent the height h. Find the exact value of each function. 31 cos 45°  cos 30°
32. 6(sin 30°)(sin 60°)
4π 11π 33. 2 sin _  3 cos _
1 2π 34. cos _ +_ sin 3π
35. (sin 45°)2 + (cos 45°)2
(cos 30°)(cos 150°) 36. __
3
H.O.T. Problems
(
6
3
)
3
sin 315°
Use HigherOrder Thinking Skills
π 37. ERROR ANALYSIS Francis and Benita are finding the exact value of cos _ . Is either 3 of them correct? Explain your reasoning.
Francis
Benita
–∏ ∏ cos _ = –cos _ 3 3
–π = cos – _ cos _ ( π + 2π) 3
3
5π = cos _
= –0.5
3
= 0.5 y
38. CHALLENGE A ray has its endpoint at the origin of
(
)
√ 3 1 the coordinate plane, and point P _ ,  _ lies on 2 2
the ray. Find the angle θ formed by the positive xaxis and the ray.
39. REASONING Is the period of a sine curve sometimes, always, or never a multiple of π? Justify your reasoning.
0
x
θ
1
( 12 ,  √23 )
40. OPEN ENDED Draw the graph of a periodic function that has a maximum value of 10 and a minimum value of 10. Describe the period of the function. 41. WRITING IN MATH Explain how to determine the period of a periodic function from its graph. Include a description of a cycle. connectED.mcgrawhill.com
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SPI 3103.3.10, SPI 3103.3.13, SPI 3102.5.2
Standardized Test Practice 44. SAT/ACT If d2 + 8 = 21, then d2  8 =
42. SHORT RESPONSE Describe the translation of the graph of f(x) = x2 to the graph of g(x) = (x + 4)2  3. 43. The rate of population decline of Hampton Cove is modeled by P(t) = 24,000e0.0064t, where t is time in years from this year and 24,000 is the current population. In how many years will the population be 10,000? A 14
B 104
C 137
F 0
H 13
G 5
J 31
K 161
45. STATISTICS If the average of three different positive integers is 65, what is the greatest possible value of one of the integers? A 192
D 375
B 193
C 194
D 195
46. GRIDDED RESPONSE If 8xy + 3 = 3, what is the value of xy?
Spiral Review Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 135) 47.
82° 14
"
"
48.
# 8
13
110°
49. #
6
#
c
$
18 118°
"
$
11
$
Determine whether each triangle has no solution, one solution, or two solutions. Then solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 134) 50. A = 72º, a = 6, b = 11
51. A = 46º, a = 10, b = 8
52. A = 110º, a = 9, b = 5
A binomial distribution has a 70% rate of success. There are 10 trials. (Lesson 127) 53. What is the probability that there will be 3 failures? 54. What is the probability that there will be at least 7 successes? 55. What is the expected number of successes? 56. GAMES The diagram shows the board for a game in which spheres are dropped down a chute. A pattern of nails and dividers causes the spheres to take various paths to the sections at the bottom. For each section, how many paths through the board lead to that section? (Lesson 116) 57. SALARIES Phillip’s current salary is $40,000 per year. His annual pay raise is always a percent of his salary at the time. What would his salary be if he got four consecutive 4% increases? (Lesson 112) Find the exact solution(s) of each system of equations. (Lesson 107) 59. 4x + y2 = 20 4x2 + y2 = 100
58. y = x + 2 y = x2
Skills Review Simplify each expression. (Lesson 14) 240 60. _ 5 1_
⎪
4
⎥
854  Lesson 136  Circular Functions
180 61. _
⎪
1 2_ 3
⎥
90 62. _ 11 ⎪2  _ 4⎥
Graphing Trigonometric Functions Then
Now
Why?
You examined periodic functions.
1
Describe and graph the sine, cosine, and tangent functions.
2
Describe and graph other trigonometric functions.
Visible light waves have different wavelengths or periods. Red has the longest wavelength and violet has the shortest wavelength.
(Lesson 136)
NewVocabulary amplitude frequency
1
Wavelength Red Wavelength Violet
Sine, Cosine, and Tangent Functions Trigonometric functions can also be
graphed on the coordinate plane. Recall that graphs of periodic functions have repeating patterns, or cycles. The horizontal length of each cycle is the period. The amplitude of the graph of a sine or cosine function equals half the difference between the maximum and minimum values of the function.
KeyConcept Sine and Cosine Functions Tennessee Curriculum Standards CLE 3103.4.3 Graph all six trigonometric functions and identify their key characteristics. ✔ 3103.4.2 Determine the period and the amplitude of a periodic function. SPI 3103.4.3 Describe and articulate the characteristics and parameters of parent trigonometric functions to solve contextual problems. Also addresses ✓3103.4.5, SPI 3103.4.2, and CLE 3103.4.5.
y = sin θ
Parent Function y
0
y
y = sin θ
1
Graph
y = cos θ
90° 180° 270° 360° 450° 540° θ
1
y = cos θ
0
90° 180° 270° 360° 450° 540° θ
1
1
Domain
{all real numbers}
{all real numbers}
Range
{y  1 ≤ y ≤ 1}
{y  1 ≤ y ≤ 1}
1
1
360°
360°
Amplitude Period
As with other functions, trigonometric functions can be transformed. For the graphs of 360° y = a sin bθ and y = a cos bθ, the amplitude = ⎪a⎥ and the period = _ . ⎪b⎥
Example 1 Find Amplitude and Period period
Find the amplitude and period of y = 4 cos 3θ. amplitude:
⎪a⎥ = ⎪4⎥ or 4
period:
360° 360° _ =_ or 120° ⎪b⎥
4
amplitude
⎪3⎥
0
GuidedPractice
y = 4 cos 3θ
y
90°
180°
270°
360° θ
4
Find the amplitude and period of each function. 1 1A. y = cos _ θ 2
1B. y = 3 sin 5θ
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StudyTip Periods In y = a sin bθ and y = a cos bθ, b represents the number of cycles in 360°. In Example 1, the 3 in y = 4 cos 3θ indicates that there are three cycles in 360°. So, there is one cycle in 120°.
Use the graphs of the parent functions to graph y = a sin bθ and y = a cos bθ. Then use the amplitude and period to draw the appropriate sine and cosine curves. You can also use θintercepts to help you graph the functions. The θintercepts of y = a sin bθ and y = a cos bθ in one cycle are as follows. y = a sin bθ
(2
1 _ (0, 0), _ · 360° , 0 b
y = a cos bθ
360° , 0) ) (_ b
360° 3 _ , 0), (_ · 360° , 0) (_14 · _ 4 b b
Example 2 Graph Sine and Cosine Functions Graph each function.
StudyTip Amplitude The graphs of y = a sin bθ and y = a cos bθ with amplitude of ⎪a⎥ have maxima at y = a and minima at y = a.
a. y = 2 sin θ Find the amplitude, the period, and the xintercepts: a = 2 and b = 1. amplitude: ⎪a⎥ = ⎪2⎥ or 2
→
The graph is stretched vertically so that the maximum value is 2 and the minimum value is 2.
360° 360° period: _ =_ or 360°
→
One cycle has a length of 360°.
⎪1⎥
⎪b⎥
xintercepts: (0, 0)
2
360° , 0) = (180°, 0) (_12 · _ b 360° , 0) = (360°, 0) (_ b
y y = 2 sin θ
1 0
90°
180°
270°
360° θ
1 2
b. y = cos 4θ
2
amplitude: ⎪a⎥ = ⎪1⎥ or 1
y y = cos 4θ
1
360° 360° period: _ =_ or 90° ⎪4⎥
⎪b⎥
0
1 _ xintercepts: _ · 360° , 0 = (22.5°, 0)
(4
b
)
90°
180°
270°
360° θ
1 2
360° , 0) = (67.5°, 0) (_34 · _ b
GuidedPractice 1 2B. y = _ sin 2θ
2A. y = 3 cos θ
2
Trigonometric functions are useful for modeling realworld periodic motion such as electromagnetic waves or sound waves. Often these waves are described using frequency. Frequency is the number of cycles in a given unit of time. The frequency of the graph of a function is the reciprocal of the period of the function. 1 So, if the period of a function is _ second, then the frequency is 100 cycles per second. 100
856  Lesson 137  Graphing Trigonometric Functions
RealWorld Example 3 Model Periodic Situations SOUND Sound that has a frequency below the human range is known as infrasound. Elephants can hear sounds in the infrasound range, with frequencies as low as 5 hertz (Hz), or 5 cycles per second. a. Find the period of the function that models the sound waves. There are 5 cycles per second, and the period is the time it takes for one cycle. 1 So, the period is _ or 0.2 second. 5
RealWorldLink Elephants are able to hear sound coming from up to 5 miles away. Humans can hear sounds with frequencies between 20 Hz and 20,000 Hz. Source: School for Champions
b. Let the amplitude equal 1 unit. Write a sine equation to represent the sound wave y as a function of time t. Then graph the equation. 2π period = _ ⎪b⎥
2π 0.2 = _ ⎪b⎥
0.2⎪b⎥ = 2π b = 10π y = a sin bθ
Write the relationship between the period and b. Substitution Multiply each side by ⎪b⎥. Multiply each side by 5; b is positive. Write the general equation for the sine function.
y = 1 sin 10πt
a = 1, b = 10π, and θ = t
y = sin 10πt
Simplify.
y
y = sin 10πt
1
0
0.1
0.2
0.4 t
0.3
1
StudyTip Amplitude and Period Note that the amplitude affects the graph along the vertical axis, and the period affects it along the horizontal axis.
GuidedPractice 3. SOUND Humans can hear sounds with frequencies as low as 20 hertz. A. Find the period of the function. B. Let the amplitude equal 1 unit. Write a cosine equation to model the sound waves. Then graph the equation.
Tangent is one of the trigonometric functions whose graphs have asymptotes.
KeyConcept Tangent Fuctions y = tan θ
Parent Function
Graph
Domain
{θ  θ ≠ 90 + 180n, n is an integer}
Range
{all real numbers}
in one cycle
1 0 90° 1
180°
Period
(2
)(
360° 1 _ (0, 0), _ · 360° , 0 , _ ,0 b
y = tan θ
2
undefined
Amplitude
θ intercepts
y
)
b
90°
270°
450°
θ
2
180° For the graph of y = a tan bθ, the period = _ , there is no amplitude, and the 180° asymptotes are odd multiples of _ .
⎪b⎥
2⎪b⎥
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StudyTip
Example 4 Graph Tangent Functions
Tangent The tangent function does not have an amplitude because it has no maximum or minimum values.
Find the period of y = tan 2θ. Then graph the function.
y = tan 2θ
180° 180° period: _ =_ or 90° ⎪b⎥
4
⎪2⎥
y
180° 180° asymptotes: _ =_ or 45° 2⎪2⎥
2⎪b⎥
Sketch asymptotes at 1 · 45° or 45°, 1 · 45° or 45°, 3 · 45° or 135°, and so on.
0
Use y = tan θ, but draw one cycle every 90°.
90°
180°
360° θ
270°
4
GuidedPractice 1 4. Find the period of y = _ tan θ. Then graph the function. 2
2
Graphs of Other Trigonometric Functions The graphs of the cosecant, secant, and cotangent functions are related to the graphs of the sine, cosine, and tangent functions.
KeyConcept Cosecant, Secant, and Cotangent Functions y = csc θ
Parent Function
y = sec θ
y = csc θ y 2
0
180°
360° θ
y 2 1
y = cos θ
y = sin θ
Graph
y = cot θ
y = sec θ
y
2
y = cot θ
O
180°
360° θ
y = tan θ
0
180°
360° θ
1 2
2
Domain
{θ  θ ≠ 180n, n is an integer}
{θ  θ ≠ 90 + 180n, n is an integer}
{θ  θ ≠ 180n, n is an integer}
Range
{y  1 > y or y > 1}
{y  1 > y or y > 1}
{all real numbers}
undefined
undefined
undefined
360°
360°
180°
Amplitude Period
StudyTip Reciprocal Functions You can use the graphs of y = sin θ, y = cos θ, and y = tan θ to graph the reciprocal functions, but these graphs are not part of the graphs of the cosecant, secant, and cotangent functions.
2
Example 5 Graph Other Trigonometric Functions Find the period of y = 2 sec θ. Then graph the function. Since 2 sec θ is a reciprocal of 2 cos θ, the graphs have the same period, 360°. The vertical asymptotes occur at the points where 2 cos θ = 0. So, the asymptotes are at θ = 90° and θ = 270°.
y = 2 sec θ 5
0
Sketch y = 2 cos θ and use it to graph y = 2 sec θ.
GuidedPractice 5. Find the period of y = csc 2θ. Then graph the function.
858  Lesson 137  Graphing Trigonometric Functions
5
y
90°
180°
270°
360° θ
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Find the amplitude and period of each function. Then graph the function.
Example 3
1. y = 4 sin θ
2. y = sin 3θ
3. y = cos 2θ
1 4. y = _ cos 3θ 2
5. SPIDERS When an insect gets caught in a spider web, the web vibrates with a frequency of 14 hertz. a. Find the period of the function. b. Let the amplitude equal 1 unit. Write a sine equation to represent the vibration of the web y as a function of time t. Then graph the equation.
Examples 4–5 Find the period of each function. Then graph the function. 6. y = 3 tan θ
7. y = 2 csc θ
8. y = cot 2θ
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Find the amplitude and period of each function. Then graph the function. 9. y = 2 cos θ
10. y = 3 sin θ
11. y = sin 2θ
12. y = cos 3θ
1 13. y = cos _ θ
14. y = sin 4θ
3 15. y = _ cos θ
3 16. y = _ sin θ
1 sin 2θ 17 y = _
18. y = 4 cos 2θ
19. y = 3 cos 2θ
2 20. y = 5 sin _ θ
4
Example 3
2
2
2
3
21. WAVES A boat on a lake bobs up and down with the waves. The difference between the lowest and highest points of the boat is 8 inches. The boat is at equilibrium when it is halfway between the lowest and highest points. Each cycle of the periodic motion lasts 3 seconds. a. Write an equation for the motion of the boat. Let h represent the height in inches and let t represent the time in seconds. Assume that the boat is at equilibrium at t = 0 seconds. b. Draw a graph showing the height of the boat as a function of time.
22. ELECTRICITY The voltage supplied by an electrical outlet is a periodic function that oscillates, or goes up and down, between 165 volts and 165 volts with a frequency of 50 cycles per second. a. Write an equation for the voltage V as a function of time t. Assume that at t = 0 seconds, the current is 165 volts. b. Graph the function. Examples 4–5 Find the period of each function. Then graph the function. 1 23. y = tan _ θ
24. y = 3 sec θ
25. y = 2 cot θ
1 26. y = csc _ θ
27. y = 2 tan θ
1 28. y = sec _ θ
2
2
3
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B
29 EARTHQUAKES A seismic station detects an earthquake wave that has a frequency of 0.5 hertz and an amplitude of 1 meter. a. Write an equation involving sine to represent the height of the wave h as a function of time t. Assume that the equilibrium point of the wave, h = 0, is halfway between the lowest and highest points. b. Graph the function. Then determine the height of the wave after 20.5 seconds. 30. PHYSICS An object is attached to a spring as shown at the right. It oscillates according to the equation y = 20 cos πt, where y is the distance in centimeters from its equilibrium position at time t. a. Describe the motion of the object by finding the following: the amplitude in centimeters, the frequency in vibrations per second, and the period in seconds. 1 second. b. Find the distance of the object from its equilibrium position at t = _ 4
c. The equation v = (20 cm)(π rad/s) · sin (π rad/s · t) represents the velocity v 1 of the object at time t. Find the velocity at t = _ second. 4
31. PIANOS A piano string vibrates at a frequency of 130 hertz. a. Write and graph an equation using cosine to model the vibration of the string y as a function of time t. Let the amplitude equal 1 unit.
C
b. Suppose the frequency of the vibration doubles. Do the amplitude and period increase, decrease, or remain the same? Explain. Find the amplitude, if it exists, and period of each function. Then graph the function. 2 32. y = 3 sin _ θ
3 1 33. y = _ cos _ θ
1 34. y = 2 tan _ θ
4 35. y = 2 sec _ θ 5
36. y = 5 csc 3θ
37. y = 2 cot 6θ
3
2
2
4
Identify the period of the graph and write an equation for each function. 38.
2
y
180°
360° θ
0 2
90° 180° 270° 360° θ
y
0
900°
1800°θ
2
4
2
4 2
2
1
H.O.T. Problems
40.
4
1 0
y
39.
4
Use HigherOrder Thinking Skills
41. CHALLENGE Describe the domain and range of y = a cos θ and y = a sec θ, where a is any positive real number. 1 1 42. REASONING Compare and contrast the graphs of y = _ sin θ and y = sin _ θ. 2
2
43. OPEN ENDED Write a trigonometric function that has an amplitude of 3 and a period of 180°. Then graph the function. 44. WRITING IN MATH Explain how to find the amplitude of y = 2 sin θ, and describe how the negative coefficient affects the graph.
860  Lesson 137  Graphing Trigonometric Functions
SPI 3103.3.4, SPI 3103.5.1, SPI 3103.3.1
Standardized Test Practice 45. SHORT RESPONSE Find the 100,001st term of the sequence.
47. Your city had a population of 312,430 ten years ago. If its current population is 418,270, by what percentage has it grown over the past 10 years?
13, 20, 27, 34, 41, …
F 25%
46. STATISTICS You bowled five games and had the following scores: 143, 171, 167, 133, and 156. What was your average? A 147
B 153
C 154
G 34%
H 66%
J 75%
48. SAT/ACT If h + 4 = b  3, then (h  2)2 =
D 156
A h2 + 4
D b2  14b + 49
B b2  6b + 3
E b2  10b + 25
C b2  18b + 81
Spiral Review Find the exact value of each expression. (Lesson 136) 49. cos 120°  sin 30°
4π π 51. 4 sin _  2 cos _
50. 3(sin 45°)(sin 60°)
3
6
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 135) 52. #
15.5 31°
53. 2 c
11.7
"
24° 5.3
54. '
3 q
17.1
16.2
21.2
( 13.6
$
) 4
A bag contains 12 blue marbles, 9 red marbles, and 8 green marbles. The marbles are drawn one at a time. Find each probability. (Lesson 123) 55. The second marble is blue, given that the first marble is green and is replaced. 56. The third marble is green, given that the first two are red and blue and not replaced. 57. The third marble is red, given that the first two are red and not replaced. 58. BANKING Rita has deposited $1000 in a bank account. At the end of each year, the bank posts interest to her account in the amount of 3% of the balance, but then takes out a $10 annual fee. (Lesson 116) a. Let b0 be the amount Rita deposited. Write a recursive equation for the balance bn in her account at the end of n years. b. Find the balance in the account after four years. Write an equation for an ellipse that satisfies each set of conditions. (Lesson 104) 59. center at (6, 3), focus at (2, 3), covertex at (6, 1) 60. foci at (2, 1) and (2, 13), covertex at (5, 7)
Skills Review Graph each function. (Lesson 57) 61. y = 2(x  3)2  4
1 62. y = _ (x + 5)2 + 2 3
63. y = 3(x + 6)2 + 7 connectED.mcgrawhill.com
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Graphing Technology Lab
Trigonometric Graphs OBJECTIVE Use a graphing calculator to transform graphs of trigonometric functions.
Tennessee Curriculum Standards
You can use a TI83/84 Plus graphing calculator to explore transformations of the graphs of trigonometric functions.
CLE 3103.4.3 Graph all six trigonometric functions and identify their key characteristics. Also addresses CLE 3103.4.5.
Activity 1 k in y = sin θ + k Graph y = sin θ, y = sin θ + 2, and y = sin θ  3 on the same coordinate plane. Describe any similarities and differences among the graphs. Use the window shown at right. Let Y1 = sin θ, Y2 = sin θ + 2, and Y3 = sin θ  3. KEYSTROKES:
[360, 360] scl: 90 by [5, 5] scl: 1
2 3 The graphs have the same shape, but different vertical positions.
Activity 2 h in y = sin (θ  h) Graph y = sin θ, y = sin (θ + 45°), and y = sin (θ  90°) on the same coordinate plane. Describe any similarities and differences among the graphs. Let Y1 = sin θ, Y2 = sin (θ + 45), and Y3 = sin (θ  90). Be sure to clear the entries from Activity 1. KEYSTROKES:
[360, 360] scl: 90 by [5, 5] scl: 1
45 90 The graphs have the same shape, but different horizontal positions.
Model and Analyze Repeat the activities for the cosine and tangent functions. 1. What is the effect of adding a constant to a trigonometric function? 2. What is the effect of adding a constant to θ in a trigonometric function? Repeat the activities for each of the following. Describe the relationship between each pair of graphs. 3. y = sin θ + 4 y = sin (2θ) + 4 5. y = 2 sin θ y = 2 sin θ  1
1 4. y = cos _ θ
(2 )
1 y = cos _ (θ + 45°) 2
6. y = cos θ  3 y = cos (θ  90°)  3
7. Write a general equation for the sine, cosine, and tangent functions after changes in amplitude a, period b, horizontal position h, and vertical position k.
862  Explore 138  Graphing Technology Lab: Trigonometric Graphs
Translations of Trigonometric Graphs Now
Why?
You translated exponential functions.
1
Graph horizontal translations of trigonometric graphs and find phase shifts.
The graphs at the right represent the waves in a bay during high and low tides. Notice that the shape of the waves does not change.
2
Graph vertical translations of trigonometric graphs.
(Lesson 81)
NewVocabulary phase shift vertical shift midline
1
h
high tide
Height (in.)
Then
low tide
0
t
Time (s)
Horizontal Translations Recall that a translation occurs when a figure is moved
from one location to another on the coordinate plane without changing its orientation. A horizontal translation of a periodic function is called a phase shift.
KeyConcept Phase Shift The phase shift of the functions y = a sin b(θ  h), y = a cos b(θ  h), and y = a tan b(θ  h) is h, where b > 0.
Words Tennessee Curriculum Standards ✔ 3103.3.4 Analyze the effect of changing various parameters on functions and their graphs. SPI 3103.3.10 Identify and/ or graph a variety of functions and their translations. SPI 3103.4.3 Describe and articulate the characteristics and parameters of parent trigonometric functions to solve contextual problems. Also addresses CLE 3103.4.3, CLE 3103.4.5, ✓3103.4.2, and ✓3103.4.5.
Models
y y = sin (θ  h), h > 0
y = sin (θ  h), h < 0 y 1
h
h θ
0
θ
0
y = sin θ
1 y = sin θ
If h < 0, the shift is ⎪h⎥ units If h > 0, the shift is h units to the left. to the right. y = cos (θ  90°) The phase shift is 90° to the right. y = tan (θ + 30°) The phase shift is 30° to the left.
Examples
The secant, cosecant, and cotangent can be graphed using the same rules.
Example 1 Graph Horizontal Translations y
State the amplitude, period, and phase shift for y = sin (θ  90°). Then graph the function. amplitude: a = 1 360° 360° period: _ =_ or 360° ⎪b⎥
1
phase shift: h = 90° Graph y = sin θ shifted 90° to the right.
y = sin (θ  90°)
1 θ 90°
0
1
90° 180° 270°360° y = sin θ
GuidedPractice 1. State the amplitude, period, and phase shift for y = 2 cos (θ + 45°). Then graph the function. connectED.mcgrawhill.com
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2
Vertical Translations Recall that the graph of y = x2 + 5 is the graph of the parent
function y = x2 shifted up 5 units. Similarly, graphs of trigonometric functions can be translated vertically through a vertical shift.
StudyTip Notation Note that sin (θ + x) ≠ sin θ + x. The first expression indicates a phase shift. The second expression indicates a vertical shift.
KeyConcept Vertical Shift The vertical shift of the functions y = a sin bθ + k, y = a cos bθ + k, and y = a tan bθ + k is k.
Words
y
Models
y
y = cos θ + k, k > 0
y = cos θ θ
0
k
k
θ 0
y = cos θ + k, k < 0 y = cos θ
If k > 0, the shift is k units up. Examples
y = sin θ + 4 y = tan θ  3
If k < 0, the shift is k units down.
The vertical shift is 4 units up. The vertical shift is 3 units down.
The secant, cosecant, and cotangent can be graphed using the same rules.
y
When a trigonometric function is shifted up or down k units, the line y = k is the new horizontal axis about which the graph oscillates. This line is called the midline, and it can be used to help draw vertical translations.
y=k
midline
90° 180° 270° 360° θ
0
y = sin θ + k
StudyTip Using Color It may be helpful to first graph the parent function in one color. Next, apply the vertical shift and graph the function in another color. Then apply the change in amplitude and graph the function in the final color.
Example 2 Graph Horizontal Translations State the amplitude, period, vertical shift, and equation of the midline for y = 1 cos θ  2. Then graph the function.
_ 2
1 amplitude: ⎪a⎥ = _
y
2
2
2π 2π period: _ =_ or 2π ⎪1⎥ ⎪b⎥
vertical shift: k = 2 midline: y = 2 1 To graph y = _ cos θ  2, first draw the
y = cos θ
y = 1 cos θ 2
θ π 2
0 2 3 4
π
π 2
3π 2
2π
y = 1 cos θ  2 2
2
1 midline. Then use it to graph y = _ cos θ 2 shifted 2 units down.
GuidedPractice 2. State the amplitude, period, vertical shift, and equation of the midline for y = tan θ + 3. Then graph the function.
864  Lesson 138  Translations of Trigonometric Graphs
y = 2
You can use the following steps to graph trigonometric functions involving phase shifts and vertical shifts.
ConceptSummary Graph Trigonometric Functions amplitude
↓
period
↓
y = a sin b(θ  h) + k ↑
↑
phase shift
vertical shift
Step 1 Determine the vertical shift, and graph the midline. Step 2 Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and minimum values of the function. Step 3 Determine the period of the function, and graph the appropriate function. Step 4 Determine the phase shift, and translate the graph accordingly.
Example 3 Graph Transformations State the amplitude, period, phase shift, and vertical shift for
_
y = 3 sin 2 (θ  π) + 4. Then graph the function. 3
amplitude: ⎪a⎥ = 3 2π 2π period: _ =_ or 3π
The period indicates that the graph will be stretched.
⎪_23 ⎥
⎪b⎥
phase shift: h = π vertical shift: k = 4 midline: y = 4
The graph will shift π to the right. The graph will shift 4 units up. The graph will oscillate around the line y = 4.
Step 1 Graph the midline. Step 2 Since the amplitude is 3, draw dashed lines 3 units above and 3 units below the midline. 2 Step 3 Graph y = 3 sin _ θ+4 3
using the midline as a reference.
y 8 7 6 5 4 3 2 1
2 y = 3 sin 3 (θ  π) + 4
π
0
StudyTip Verifying a Graph After drawing the graph of a trigonometric function, select values of θ and evaluate them in the equation to verify your graph.
Step 4 Shift the graph π to the right. CHECK
2
y = 3 sin 3 θ + 4
y=4
π
2π
3π
4π θ
You can check the accuracy of your transformation by evaluating the function for various values of θ and confirming their location on the graph.
GuidedPractice 3. State the amplitude, period, phase shift, and vertical shift for π 1 y = 2 cos _ θ+_  2. Then graph the function. 2
(
2
)
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The sine wave occurs often in physics, signal processing, music, electrical engineering, and many other fields.
RealWorld Example 4 Represent Periodic Functions WAVE POOL The height of water in a wave pool oscillates between a maximum of 13 feet and a minimum of 5 feet. The wave generator pumps 6 waves per minute. Write a sine function that represents the height of the water at time t seconds. Then graph the function. Step 1 Write the equation for the midline, and determine the vertical shift. y = _ or 9 13 + 5 2
The midline lies halfway between the maximum and minimum values.
Since the midline is y = 9, the vertical shift is k = 9.
RealWorldLink In some wave pools, surfers can ride waves up to 70 meters. Source: Orlando Wave Pool
Step 2 Find the amplitude. ⎪a⎥
= ⎪13  9⎥ or 4
So, a = 4.
Find the difference between the midline value and the maximum value.
Step 3 Find the period. Since there are 6 waves per minute, there is 1 wave every 10 seconds. So, the period is 10 seconds. 2π 10 = _ ⎪b⎥
2π ⎪b⎥ = _ 10 π b = ±_ 5
_
period = 2π ⎪b⎥
Solve for ⎪b⎥. Simplify.
WatchOut! Parent Functions Often the graph of a trigonometric function can be represented by more than one equation. For example, the graphs of y = cos θ and y = sin (θ + 90°) are the same.
Step 4 Write an equation for the function. h = a sin b(t  h) + k
Write the equation for sine relating height h and time t.
π h = 4 sin _ (t  0) + 9
π Substitution: a = 4, b = _ , h = 0, k = 9
5 π h = 4 sin _ t+9 5
5
Simplify.
Then graph the function.
π
h
h = 4 sin 5 t + 9
14 12 10 8 6 4 2 0
5
10
15
20
25 t
GuidedPractice 4. WAVE POOL The height of water in a wave pool oscillates between a maximum of 14 feet and a minimum of 6 feet. The wave generator pumps 5 waves per minute. Write a cosine function that represents the height of water at time t seconds. Then graph the function.
866  Lesson 138  Translations of Trigonometric Graphs
Check Your Understanding Example 1
State the amplitude, period, and phase shift for each function. Then graph the function. 1. y = sin (θ  180°)
π 2. y = tan θ  _
π 3. y = sin θ  _
1 4. y = _ cos (θ + 90°)
(
Example 2
2
)
4
)
State the amplitude, period, vertical shift, and equation of the midline for each function. Then graph the function. 5. y = cos θ + 4
6. y = sin θ  2
1 7. y = _ tan θ + 1
8. y = sec θ  5
State the amplitude, period, phase shift, and vertical shift for each function. Then graph the function. 9. y = 2 sin (θ + 45°) + 1 1 11. y = _ tan 2(θ + 30°) + 3 4
Example 4
(
2
2
Example 3
= StepbyStep Solutions begin on page R20.
10. y = cos 3(θ  π)  4 π 1 12. y = 4 sin _ θ_ +5 2
(
2
)
13. EXERCISE While doing some moderate physical activity, a person’s blood pressure oscillates between a maximum of 130 and a minimum of 90. The person’s heart rate is 90 beats per minute. Write a sine function that represents the person’s blood pressure P at time t seconds. Then graph the function.
Practice and Problem Solving Example 1
State the amplitude, period, and phase shift for each function. Then graph the function. 14. y = cos (θ + 180°)
15. y = tan (θ  90°)
16. y = sin (θ + π)
π 17. y = 2 sin θ + _
1 18. y = tan _ (θ + 30°) 2
Example 2
20. y = cos θ + 3
21. y = tan θ  1
1 22. y = tan θ + _
23 y = 2 cos θ  5
24. y = 2 sin θ  4
1 25. y = _ sin θ + 7 3
State the amplitude, period, phase shift, and vertical shift for each function. Then graph the function. 26. y = 4 sin (θ  60°)  1
1 27. y = cos _ (θ  90°) + 2
28. y = tan (θ + 30°)  2
π 29. y = 2 tan 2 θ + _ 5
π 1 30. y = _ sin θ  _ +4
1 31. y = cos 3(θ  45°) + _
32. y = 3 + 5 sin 2(θ  π)
π 1 33. y = 2 + 3 sin _ θ_ 3( 2)
2
Example 4
( 2) π 19. y = 3 cos (θ  _ 3)
State the amplitude, period, vertical shift, and equation of the midline for each function. Then graph the function.
2
Example 3
Extra Practice begins on page 947.
(
2
)
2
(
4
)
2
34. TIDES The height of the water in a harbor rose to a maximum height of 15 feet at 6:00 p.m. and then dropped to a minimum level of 3 feet by 3:00 a.m. The water level can be modeled by the sine function. Write an equation that represents the height h of the water t hours after noon on the first day. connectED.mcgrawhill.com
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35. LAKES A buoy marking the swimming area in a lake oscillates each time a speed boat goes by. Its distance d in feet from the bottom of the lake is given by 3π d = 1.8 sin _ t + 12, where t is the time in seconds. Graph the function. Describe 4 the minimum and maximum distances of the buoy from the bottom of the lake when a boat passes by.
36. FERRIS WHEEL Suppose a Ferris wheel has a diameter of approximately 520 feet and makes one complete revolution in 30 minutes. Suppose the lowest car on the Ferris wheel is 5 feet from the ground. Let the height at the top of the wheel represent the height at time 0. Write an equation for the height of a car h as a function of time t minutes. Then graph the function. Write an equation for each translation. 37. y = sin x, 4 units to the right and 3 units up 38. y = cos x, 5 units to the left and 2 units down 39. y = tan x, π units to the right and 2.5 units up
B
40. JUMP ROPE The graph at the right approximates the height of a jump rope h in inches as a function of time t in seconds. A maximum point on the graph is (1.25, 68), and a minimum point is (2.75, 2).
80 70 60 50 40 30 20 10
a. Describe what the maximum and minimum points mean in the context of the situation. b. What is the equation for the midline, the amplitude, and the period of the function?
h
0
1
2
3
4t
c. Write an equation for the function. 41 CAROUSEL A horse on a carousel goes up and down 3 times as the carousel makes one complete rotation. The maximum height of the horse is 55 inches, and the minimum height is 37 inches. The carousel rotates once every 21 seconds. Assume that the horse starts and stops at its median height. a. Write an equation to represent the height of the horse h as a function of time t seconds. b. Graph the function. c. Use your graph to estimate the height of the horse after 8 seconds. Then use a calculator to find the height to the nearest tenth. 42. TEMPERATURES During one month, the outside temperature fluctuates between 40°F and 50°F. A cosine curve approximates the change in temperature, with a high of 50°F being reached every four days. a. Describe the amplitude, period, and midline of the function that approximates the temperature y on day d. b. Write a cosine function to estimate the temperature y on day d. c. Sketch a graph of the function. d. Estimate the temperature on the 7th day of the month. Find a coordinate that represents a maximum for each graph. π 43. y = 2 cos x  _ )
π 44. y = 4 sin x + _ )
π 45. y = 3 tan x + _ +2 2
π 46. y = 3 sin x  _ 4
(
(
2
)
868  Lesson 138  Translations of Trigonometric Graphs
(
3
(
4
)
Compare each pair of graphs. 47. y = cos 3θ and y = sin 3(θ  90°) 48. y = 2 + 0.5 tan θ and y = 2 + 0.5 tan (θ + π) 5π π 49. y = 2 sin θ  _ and y = 2 sin θ + _
(
C
6
(
)
6
)
Identify the period of each function. Then write an equation for the graph using the given trigonometric function. 50. sine
y
2 0 90 2
51. cosine
90
2
180 270 360 θ
0 90 2
4 6
52. cosine
90
180 270 360 θ
90
180 270 360 θ
4
y
6
y
4
53 sine
4
4
2
2 90 180 270 360 θ
0 90 2
y
6
0 90 2 4
4
State the period, phase shift, and vertical shift. Then graph the function. 54. y = csc (θ + π)
55. y = cot θ + 6
π 56. y = cot θ  _ 2
1 57. y = _ csc 3(θ  45°) + 1
1 58. y = 2 sec _ (θ  90°)
π 59. y = 4 sec 2 θ + _ 3
(
6
)
2
H.O.T. Problems
2
(
2
)
Use HigherOrder Thinking Skills
60. CHALLENGE If you are given the amplitude and period of a cosine function, is it sometimes, always, or never possible to find the maximum and minimum values of the function? Explain your reasoning. 61. REASONING Describe how the graph of y = 3 sin 2θ + 1 is different from y = sin θ. 62. WRITING IN MATH Describe two different phase shifts that will translate the sine curve onto the cosine curve shown at the right. Then write an equation for the new sine curve using each phase shift. 63. OPEN ENDED Write a periodic function that has an amplitude of 2 and midline at y = 3. Then graph the function. 64. REASONING How many different sine graphs pass through the origin (nπ, 0)? Explain your reasoning.
y = cos θ
y 1
90
0
1
90
180 270 360
θ
y = sin θ
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SPI 3103.3.3, SPI 3103.3.1, SPI 3102.2.1, SPI 3108.4.3
Standardized Test Practice 3x  1 65. GRIDDED RESPONSE The expression _ +_ 4 4 is how much greater than x? x+6
66. Expand (a  b)4.
F 7 G 0, 7
H 7, 13 J no solution
68. GEOMETRY Using the figures below, what is the average of a, b, c, d, and f?
a4  b4 a4  4ab + b4 a4 + 4a3b + 6a2b2 + 4ab3 + b4 a4  4a3b + 6a2b2  4ab3 + b4
A B C D
67. Solve √ x  3 + √ x + 2 = 5.
b° a°
A 21
f° c°
B 45
d°
C 50
D 54
Spiral Review Find the amplitude and period of each function. Then graph the function. (Lesson 137) 69. y = 2 cos θ
70. y = 3 sin θ
71. y = sin 2θ
Find the exact value of each expression. (Lesson 136) 4π 72. sin _ 3
73. sin (30°)
74. cos 405°
State whether each situation represents an experiment or an observational study. If it is an experiment, identify the control group and the treatment group. Then determine whether there is bias. (Lesson 121) 75. Find 220 people and randomly split them into two groups. One group exercises for an hour a day and the other group does not. Then compare their body mass indexes. 76. Find 200 students, half of whom play soccer, and compare the amounts of time spent sleeping. 77. Find 100 students, half of whom have parttime jobs, and compare their grades. #
78. GEOMETRY Equilateral triangle ABC has a perimeter of 39 centimeters. If the midpoints of the sides are connected, a smaller equilateral triangle results. Suppose the process of connecting midpoints of sides and drawing new triangles is continued indefinitely. (Lesson 114) a. Write an infinite geometric series to represent the sum of the perimeters of all of the triangles. b. Find the sum of the perimeters of all of the triangles.
"
79. CONSTRUCTION A construction company will be fined for each day it is late completing a bridge. The daily fine will be $4000 for the first day and will increase by $1000 each day. Based on its budget, the company can only afford $60,000 in total fines. What is the maximum number of days it can be late? (Lesson 113)
Skills Review Find each value of θ. Round to the nearest degree. (Lesson 131) 7 80. sin θ = _
8 4 83. cos θ = _ 5
9 81. tan θ = _
10 5 _ 84. sin θ = 6
870  Lesson 138  Translations of Trigonometric Graphs
1 82. cos θ = _ 4
2 85. tan θ = _ 7
$
Inverse Trigonometric Functions Then
Now
Why?
You graphed trigonometric functions.
1
Find values of inverse trigonometric functions.
2
Solve equations by using inverse trigonometric functions.
The leaning bookshelf at the right is 15 inches from the wall and reaches a height of 75 inches. In Lesson 131, you learned how to use the inverse of a trigonometric function to find the measure of acute angle θ.
(Lesson 137)
15 or 0.2 tan θ = _ 75
θ
75 in.
Use the tangent function.
Find an angle that has a tangent of 0.2.
[TAN1] .2
11.30993247
So, the measure of θ is about 11°. 15 in.
NewVocabulary principal values Arcsine function Arccosine function Arctangent function
Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems. ✔ 3103.4.6 Know and be able to use the fundamental trigonometric identities.
1
Inverse Trigonometric Functions If you know the value of
2π
a trigonometric function for an angle, you can use the inverse to find the angle. Recall that an inverse function is the relation in which all values of x and y are reversed. The inverse of y = sin x, x = sin y, is graphed at the right. Notice that the inverse is not a function because there are many values of y for each value of x. If you restrict the π π domain of the sine function so that _ ≤x≤_ , then 2 2 the inverse is a function.
y
3π 2
x = sin y
π π 2
1.0
0 π
1.0
x
2
π  3π 2
2π
The values in this restricted domain are called principal values. Trigonometric functions with restricted domains are indicated with capital letters. π π • y = Sin x if and only if y = sin x and _ ≤x≤_ . 2
2
• y = Cos x if and only if y = cos x and 0 ≤ x ≤ π. π π • y = Tan x if and only if y = tan x and _ ≤x≤_ . 2
2
You can use functions with restricted domains to define inverse trigonometric functions. The inverses of the sine, cosine, and tangent functions are the Arcsine, Arccosine, and Arctangent functions, respectively.
KeyConcept Inverse Trigonometric Functions Inverse Function
Arcsine
Arccosine
Arctangent
Symbols
Domain
y = Arcsin x y = Sin1 x
1 ≤ x ≤ 1
y = Arccos x y = Cos1 x
1 ≤ x ≤ 1
y = Arctan x y = Tan1 x
all real numbers
Range
Model
π π _ ≤y≤_ 2
π 2
2
90° ≤ y ≤ 90° 0≤y≤π 0° ≤ y ≤ 180° π π _ ≤y≤_ 2
1  1
0
2
π
y
y = sin1 x
1 2
1
x
2
2
90° ≤ y ≤ 90° connectED.mcgrawhill.com
871
ReviewVocabulary inverse functions If f and f1 are inverse functions, then f(a) = b if and only if f1(b) = a. (Lesson 72)
1 In the relation y = cos1 x, if x = _ , y = 60°, 300°, and all angles that are coterminal with 2
1 those angles. In the function y = Cos1 x, if x = _ , y = 60° only. 2
Example 1 Evaluate Inverse Trigonometric Functions Find each value. Write angle measures in degrees and radians.
( _2 )
a. Cos1  1
1 Find the angle θ for 0° ≤ θ ≤ 180° that has a cosine value of _ . 2
Method 1 Use a unit circle.
( 12 , √23 )
Find a point on the unit circle that
1
2π
1 has an xcoordinate of _ . 2
θ
1 When θ = 120°, cos θ = _ .
Angle Measure Remember that when evaluating an inverse trigonometric function, the result is an angle measure.
120° = 3 1 x
0
1
2 2π 1 So, Cos1 _ = 120° or _. 3 2
( )
StudyTip
y
1
Method 2 Use a calculator. [COS 1]
KEYSTROKES:
1
2
120
2π 1 Therefore, Cos1 _ = 120° or _ .
( 2)
3
b. Arctan 1 Find the angle θ for 90° ≤ θ ≤ 90° that has a tangent value of 1. [TAN 1] 1
KEYSTROKES:
45
π Therefore, Arctan 1 = 45° or _ . 4
GuidedPractice
(
√ 2 2
1B. Arcsin _
1A. Cos1 0
)
When finding a value when there are multiple trigonometric functions involved, use the order of operations to solve.
Example 2 Find Trigonometric Value
_)
(
Find tan Cos1 1 . Round to the nearest hundredth. 2
Use a calculator. [COS 1] 1
KEYSTROKES:
2
1.732050808
1 So, tan Cos1 _ ≈ 1.73.
(
2
)
1 CHECK Cos1 _ = 60° and tan 60° ≈ 1.73. So, the answer is correct. 2
GuidedPractice Find each value. Round to the nearest hundredth. 3 2A. sin Tan1 _
(
8
)
872  Lesson 139  Inverse Trigonometric Functions
(
√ 2 2
2B. cos Arccos _
)
2
Solve Equations by Using Inverses You can rewrite trigonometric equations to solve for the measure of an angle. SPI 3.5
Test Example 3 If Sin θ = 0.35, find θ.
TestTakingTip Eliminate Possibilities The function Sin restricts the possible angle measures to Quadrants I or IV. Because 0.35 is negative, look for an angle measure in Quadrant IV.
A 20.5°
B 0.6°
C 0.6°
D 20.5°
Read the Test Item The sine of angle θ is 0.35. This can be written as Arcsin (0.35) = θ. Solve the Test Item Use a calculator. [SIN 1]
KEYSTROKES:
.35
20.48731511
So, θ ≈ 20.5°. The answer is A.
GuidedPractice 3. If Tan θ = 1.8, find θ. F 0.03°
G 29.1°
H 60.9°
J
no solution
Inverse trigonometric functions can be used to determine angles of inclination, depression, and elevation.
RealWorld Example 4 Use Inverse Trigonometric Functions WATER SKIING A water ski ramp is 6 feet tall and 9 feet long, as shown at the right. Write an inverse trigonometric function that can be used to find θ, the angle the ramp makes with the water. Then find the measure of the angle. Round to the nearest tenth.
9 ft
6 ft
θ
Because the measures of the opposite side and the hypotenuse are known, you can use the sine function. 6 sin θ = _
Sine function
9
6 θ = Sin1 _
Inverse sine function
θ ≈ 41.8°
Use a calculator.
9
RealWorldCareer Sport Science Administrator A sport science administrator provides sport science information to players, coaches, and parents. He or she implements testing, training, and treatment programs for athletes. A master’s degree in sport science or a related area is recommended.
So, the angle of the ramp is about 41.8°. CHECK
6 Using your calculator, sin 41.8 ≈ 0.66653 ≈ _ . 9 So, the answer is correct.
GuidedPractice 4. SKIING A ski trail is shown at the right. Write an inverse trigonometric function that can be used to find θ, the angle the trail makes with the ground in the valley. Then find the angle. Round to the nearest tenth.
5 ft θ 12 ft
connectED.mcgrawhill.com
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Check Your Understanding Example 1
= StepbyStep Solutions begin on page R20.
Find each value. Write angle measures in degrees and radians. 1 1. Sin1 _
2. Arctan ( √ 3)
2
Example 2
Find each value. Round to the nearest hundredth if necessary. 4 4. cos Arcsin _
(
Example 3
3. Arccos (1)
5
)
(
√ 3 2
6. sin Sin1 _
5. tan (Cos1 1)
)
7. MULTIPLE CHOICE If Sin θ = 0.422, find θ. A 25°
B 42°
C 48°
D 65°
Solve each equation. Round to the nearest tenth if necessary. 8. Cos θ = 0.9 Example 4
9. Sin θ = 0.46
10. Tan θ = 2.1
11. SNOWBOARDING A cross section of a superpipe for snowboarders is shown at the right. Write an inverse trigonometric function that can be used to find θ, the angle that describes the steepness of the superpipe. Then find the angle to the nearest degree.
θ 18 ft
6.2 ft
Practice and Problem Solving Example 1
Find each value. Write angle measures in degrees and radians.
(2)
Example 2
Example 3
Example 4
Extra Practice begins on page 947.
(2) (_23 )
√ 3 12. Arcsin _
√ 3 13 Arccos _
15. Tan1 √ 3
16. Cos1
√
14. Sin1 (1)
(
√ 3 3
17. Arctan _
)
Find each value. Round to the nearest hundredth if necessary. 18. tan (Cos1 1)
1 ⎤ 19. tan ⎡Arcsin _
3 20. cos Tan1 _
21. sin (Arctan √ 3)
4 22. cos Sin1 _
23. sin ⎢Cos1 _
⎣
(
( 2 )⎦
9
)
(
5
⎡ ⎣
(
) √ 2 2
)⎤⎦
Solve each equation. Round to the nearest tenth if necessary. 24. Tan θ = 3.8
25. Sin θ = 0.9
26. Sin θ = 2.5
27. Cos θ = 0.25
28. Cos θ = 0.56
29. Tan θ = 0.2
30. BOATS A boat is traveling west to cross a river that is 190 meters wide. Because of the current, the boat lands at point Q, which is 59 meters from its original destination point P. Write an inverse trigonometric function that can be used to find θ, the angle at which the boat veered south of the horizontal line. Then find the measure of the angle to the nearest tenth.
874  Lesson 139  Inverse Trigonometric Functions
/ 1 59 m 2
190 m Ĥ
31. TREES A 24foot tree is leaning 2.5 feet left of vertical, as shown in the figure. Write an inverse trigonometric function that can be used to find θ, the angle at which the tree is leaning. Then find the measure of the angle to the nearest degree.
B
2.5 ft
24 ft
Ĥ
32. DRIVING An expressway offramp curve has a radius of 52 meters and is designed for vehicles to safely travel at speeds up to 45 kilometers per hour (or 12.5 meters per second). The equation below represents the angle θ of the curve. What is the measure of the angle to the nearest degree? (12.5 ms)2
tan θ = __ 2 (52 m)(9.8 ms )
33 TRACK AND FIELD A shotputter throws the shot with an initial speed of 15 meters 15 ms (sin x)
represents the time in seconds at which the per second. The expression __ 2 9.8 ms
shot reached its maximum height. In the expression, x is the angle at which the shot was thrown. If the maximum height of the shot was reached in 1.0 second, at what angle was it thrown? Round to the nearest tenth. Solve each equation for 0 ≤ θ ≤ 2π. 34. csc θ = 1
35. sec θ = 1
36. sec θ = 1
1 37. csc θ = _ 2
38. cot θ = 1
39. sec θ = 2
MULTIPLE REPRESENTATIONS Consider y = Cos1 x.
40.
a. Graphical Sketch a graph of the function. Describe the domain and the range. b. Symbolic Write the function using different notation. c. Numerical Choose a value for x between 1 and 0. Then evaluate the inverse cosine function. Round to the nearest tenth. d. Analytical Compare the graphs of y = cos x and y = Cos1 x.
H.O.T. Problems C
Use HigherOrder Thinking Skills
41. CHALLENGE Determine whether cos (Arccos x) = x for all values of x is true or false. If false, give a counterexample. 42. ERROR ANALYSIS Desiree and Oscar are solving cos θ = 0.3 where 90 < θ < 180. Is either of them correct? Explain your reasoning.
Desiree cos θ = 0.3 –1 cos 0.3 = 162.5°
Oscar cos θ = 0.3 cos–1 0.3 = 72.5º
43. REASONING Explain how the domain of y = Sin1 x is related to the range of y = Sin x. 44. OPEN ENDED Write an equation with an Arcsine function and an equation with a Sine function that both involve the same angle measure. 45. WRITING IN MATH Compare and contrast the relations y = tan1 x and y = Tan1 x. Include information about the domains and ranges. 46. REASONING Explain how Sin1 8 and Cos1 8 are undefined while Tan1 8 is defined. connectED.mcgrawhill.com
875
SPI 3103.3.3, SPI 3103.3.11, SPI 3103.3.6, SPI 3102.2.1
Standardized Test Practice _2 + 2
x . 47. Simplify _ 2
49. If f(x) = 2x2  3x and g(x) = 4  2x, what is g[f(x)]?
_2 x
1+x A _
1x C _
2 B _ x
D x
F G H J
1+x
1x
48. SHORT RESPONSE What is the equation of the graph below? 2
y
−20
g[f(x)] = 4 + 6x  8x2 g[f(x)] = 4 + 6x  4x2 g[f(x)] = 20  26x + 8x2 g[f(x)] = 44  38x + 8x2
50. If g is a positive number, which of the following is equal to 12g?
2 4 6 8 10x
−4 −6 −8 −10
A
144g √
B
12g2 √
C
24g2 √
D 6 √ 4g2
Spiral Review 51. RIDES The Cosmoclock 21 is a huge Ferris wheel in Japan. The diameter is 328 feet. Suppose a rider enters the ride at 0 feet, and then rotates in 90° increments counterclockwise. The table shows the angle measures of rotation and the height in feet above the ground of the rider. (Lesson 138)
328 ft
a. A function that models the data is y = 164 · [sin (x  90°)] + 164. Identify the vertical shift, amplitude, period, and phase shift of the graph. b. Write an equation using the sine that models the position of a rider on the Vienna Giant Ferris Wheel in Austria, with a diameter of 200 feet. Check your equation by plotting the points and the equation with a graphing calculator.
164 ft 90°
Angle
Height
Angle
Height
0° 90° 180° 270° 360°
0 164 328 164 0
450° 540° 630° 720°
164 328 164 0
52. TIDES The world’s record for the highest tide is held by the Minas Basin in Nova Scotia, Canada, with a tidal range of 54.6 feet. A tide is at equilibrium when it is at its normal level halfway between its highest and lowest points. Write an equation to represent the height h of the tide. Assume that the tide is at equilibrium at t = 0, that the high tide is beginning, and that the tide completes one cycle in 12 hours. (Lesson 137) Solve each equation. (Lesson 84) 53. log3 5 + log3 x = log3 10
54. log4 a + log4 9 = log4 27
55. log10 16  log10 2t = log10 2
56. log7 24  log7 (y + 5) = log3 8
Skills Review Find the exact value of each trigonometric function. (Lesson 133) 57. cos 3π
58. tan 120°
876  Lesson 139  Inverse Trigonometric Functions
59. sin 300°
7π 60. sec _ 6
Study Guide and Review Study Guide KeyConcepts
KeyVocabulary ambiguous case (p. 834)
period (p. 849)
amplitude (p. 855)
periodic function (p. 849)
angle of depression (p. 812)
phase shift (p. 863)
angle of elevation (p. 812)
principle values (p. 871)
Arccosine function (p. 871)
quadrantal angle (p. 826)
Arcsine function (p. 871)
radian (p. 819)
Arctangent function (p. 871)
reference angle (p. 826)
central angle (p. 820)
secant (p. 808)
circular function (p. 848)
sine (p. 808)
cosecant (p. 808)
solving a triangle (p. 833)
Law of Sines and Law of Cosines (Lessons 134 and 135)
cosine (p. 808)
standard position (p. 817)
sin A sin B sin C • _ =_ =_
cotangent (p. 808)
tangent (p. 808)
• a 2 = b 2 + c 2  2bc cos A b 2 = a 2 + c 2  2ac cos B c 2 = a 2 + b 2  2ab cos C
coterminal angles (p. 818)
terminal side (p. 817)
cycle (p. 849)
trigonometric function (p. 808)
frequency (p. 856)
trigonometric ratio (p. 808)
Circular and Inverse Trigonometric Functions
initial side (p. 817)
trigonometry (p. 808)
(Lessons 136 and 139)
Law of Cosines (p. 841)
unit circle (p. 848)
Law of Sines (p. 833)
vertical shift (p. 864)
Right Triangle Trigonometry (Lesson 131) adj opp opp hyp hyp adj adj hyp hyp _ _ _ csc θ = opp , sec θ = , cot θ = opp adj
• sin θ = _, cos θ = _, tan θ = _,
Angle Measures and Trigonometric Functions of General Angles (Lessons 132 and 133) • The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. • You can find the exact values of the six trigonometric functions of θ, given the coordinates of a point P (x, y ) on the terminal side of the angle.
a
b
c
• If the terminal side of an angle θ in standard position intersects the unit circle at P (x, y ), then cos θ = x and sin θ = y. π π • y = Sin x if y = sin x and _ ≤x≤_
Graphing Trigonometric Functions (Lesson 137) • For trigonometric functions of the form y = a sin bθ and y = a cos bθ, the amplitude is ⎪a⎥, and the period is 360° _ _ or 2π . ⎪b⎥
midline (p. 864)
2
2
VocabularyCheck State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.
b
180° _ • The period of y = a tan bθ is _ or π . ⎪b⎥
⎪b⎥
Be sure the Key Concepts are noted in your Foldable.
Trignonometric Functions
StudyOrganizer
1. The Law of Cosines is used to solve a triangle when two angles and any sides are known. 2. An angle on the coordinate plane is in standard position if the vertex is at the origin and one ray is on the positive xaxis. 3. Coterminal angles are angles in standard position that have the same terminal side.
131
4. A horizontal translation of a periodic function is called a phase shift. 5. The inverse of the sine function is the cosecant function. 6. The cycle of the graph of a sine or cosine function equals half the difference between the maximum and minimum values of the function. connectED.mcgrawhill.com
877
Study Guide and Review Continued LessonbyLesson Review
CLE 3103.4.5
1311Right Triangle Trigonometry
(pp. 808–816)
Solve ABC by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
"
Example 1 c
b
7. c = 12, b = 5
Solve ABC by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
a
$
#
8. a = 10, B = 55°
Find b.
9. B = 75°, b = 15 10. B = 45°, c = 16 11. A = 35°, c = 22
Find A.
2 12. sin A = _ ,a=6
# 9
a2 + b2 = c2 9 2 + b 2 = 16 2 b = √ 16 2  9 2 b ≈ 13.2
$
16
b
"
9 sin A = _ 16
Use a calculator. To the nearest degree, A = 34°.
3
13. TRUCK The back of a moving truck is 3 feet off of the ground. What length does a ramp off the back of the truck need to be in order for the angle of elevation of the ramp to be 20°?
Find B.
34° + B ≈ 90° B ≈ 56°
Therefore, b ≈ 13.2, A ≈ 34°, and B ≈ 56°.
CLE 3103.4.5, ✔3103.4.1, ✔3103.4.4
1322 Angles and Angle Measures
(pp. 817–823)
Rewrite each degree measure in radians and each radian measure in degrees. 14. 215°
5π 15. _
16. 3π
17. 315°
Example 2 Rewrite 160° in radians.
(
180°
160π 8π radians or _ =_ 180
Find one angle with positive measure and one angle with negative measure coterminal with each angle. 18. 265°
19. 65°
)
π radians 160° = 160° _
2
7π 20. _ 2
21. BICYCLE A bicycle tire makes 8 revolutions in one minute. The tire has a radius of 15 inches. Find the angle θ in radians through which the tire rotates in one second.
878  Chapter 13  Study Guide and Review
9
Example 3 Find one angle with positive measure and one angle with negative measure coterminal with 150°. positive angle: 150° + 360° = 510°
Add 360°.
negative angle: 150°  360° = 210°
Subtract 360°.
CLE 3103.4.5
1333 Trigonometric Functions of General Angles
(pp. 825–831)
Find the exact value of each trigonometric function.
Example 4
22. cos 135°
23. tan 150°
Find the exact value of sin 120°.
24. sin 2π
3π 25. cos _
Because the terminal side of 120° lies in Quadrant II, the reference angle θ is 180°  120° or 60°. The sine function is positive in Quadrant II,
2
The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ. 26. P (4, 3)
y θ = 120°
θ' = 60°
x
0
√ 3 2
so sin 120° = sin 60° or _.
27. P (5, 12)
Example 5
28. P (16, 12) 29. BALL A ball is thrown off the edge of a building at an angle of 70° and with an initial velocity of 5 meters per second. The equation that represents the horizontal distance of the ball x is x = v 0 (cos θ)t, where v 0 is the initial velocity, θ is the angle at which it is thrown, and t is the time in seconds. About how far will the ball travel in 10 seconds?
The terminal side of θ in standard position contains the point (6, 5). Find the exact values of the six trigonometric functions of θ. sin θ =
cos θ =
tan θ =
5 √ 61 _y or _
6 √ 61 _x or _
_y or _5
r
61
r
x
61
6
csc θ =
sec θ =
cot θ =
√ 61 _r or _
√ 61 _r or _
_x or _6
y
5
x
y
6
5
CLE 3103.4.4, CLE 3103.4.5
1344 Law of Sines
(pp. 832–839)
Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 30. C = 118°, c = 10, a = 4
Example 6 First, find the measure of the third angle.
32. A = 70°, a = 5, c = 16 33. BOAT Kira and Mallory are standing on opposite sides of a river. How far is Kira from the boat? Round to the nearest tenth if necessary.
Now use the Law of Sines to find a and c. Write two equations, each with one variable. sin C sin B _ =_
30° 85°
8
60° + 70° + a = 180° A = 50°
31. A = 25°, a = 15, c = 18
.BMMPSZ
"
Solve ABC.
90 ft
,JSB
60°
70°
#
$
sin B sin A _ =_
c b sin 60° sin 70° _ =_ c 8 8 sin 70° c=_ sin 60°
a b sin 60° sin 50° _ =_ a 8 8 sin 50° a=_ sin 60°
c ≈ 8.7
a ≈ 7.1
Therefore, A = 50°, c ≈ 8.7, and a ≈ 7.1.
connectED.mcgrawhill.com
879
Study Guide and Review Continued CLE 3103.4.5
1355 Law of Cosines
(pp. 841–846)
Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 34.
35.
#
#
15
16 21
"
15
12
"
Solve ABC for C = 55°, b = 11, and a = 18. You are given the measure of two sides and the included angle. Begin by drawing a diagram and using the Law of Cosines to determine c.
#
18
c 2 = a 2 + b 2  2ab cos C
80°
$
Example 7
c 2 = 182 + 112  2(18)(11) cos 55° $
55°
"
2
c ≈ 217.9
11
$
c ≈ 14.8
36. C = 75°, a = 5, b = 7
Next, you can use the Law of Sines to find the measure of angle A.
37. A = 42°, a = 9, b = 13 38. b = 8.2, c = 15.4, A = 35°
sin A _ _ ≈ sin 55°
39. FARMING A farmer wants to fence a piece of his land. Two sides of the triangular field have lengths of 120 feet and 325 feet. The measure of the angle between those sides is 70°. How much fencing will the farmer need?
18
14.8
18 sin 55° sin A ≈ _ or A is about 85.0° 14.8
The measure of the angle B is approximately 180  (85.0 + 55) or 40.0°. Therefore, c ≈ 14.8, A ≈ 85.0°, and B ≈ 40.0°.
✔3103.1.7, CLE 3103.4.1, ✔3103.4.3, SPI 3103.4.1
1366 Circular Functions
(pp. 848–854)
Find the exact value of each function.
Example 8
40. cos (210°)
41. (cos 45°)(cos 210°)
Find the exact value of sin 510°.
7π 42. sin _
π π 43. cos _ sin _
4
(
2 )(
2)
44. Determine the period of the function. y
= sin 150° 1 =_ 2
Example 9
1
0
sin 510° = sin (360° + 150°)
Determine the period of the function below. 1 2 3 4 5 6 7 8 9 101112 x
1
y
1 0
45. A wheel with a diameter of 18 inches completes 4 revolutions in 1 minute. What is the period of the function that describes the height of one spot on the outside edge of the wheel as a function of time?
π
3π 2
2π θ
1
π The pattern repeats itself at _ , π, and so on. So, the 2 π _ period is . 2
880  Chapter 13  Study Guide and Review
π 2
CLE 3103.4.3, ✔3103.4.2, SPI 3103.4.3
1377 Graphing Trigonometric Functions
(pp. 855–861)
Find the amplitude, if it exists, and period of each function. Then graph the function. 46. y = 4 sin 2θ 1 47. y = cos _ θ
Example 10 Find the amplitude and period of y = 2 cos 4θ. Then graph the function. amplitude: ⎪a⎥ = ⎪2⎥ or 2. The graph is stretched vertically so that the maximum value is 2 and the minimum value is 2.
2
48. y = 3 csc θ
y
period:
49. y = 3 sec θ
2
360° _ _ = 360° or 90° ⎪4⎥ ⎪b⎥
50. y = tan 2θ
1
1 51. y = 2 csc _ θ
180° 90°
2
52. When Lauren jumps on a trampoline it vibrates with a frequency of 10 hertz. Let the amplitude equal 5 feet. Write a sine equation to represent the vibration of the trampoline y as a function of time t.
0 1
90°
180°θ
2
✔3103.3.4, SPI 3103.3.10, SPI 3103.4.3
1388 Translations of Trigonometric Graphs
(pp. 863–870)
Example 11
State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.
State the vertical shift, amplitude, period, and phase shift
53. y = 3 sin [2(θ  90°)] + 1
of y = 2 sin 3 θ + π
1 54. y = _ tan [2(θ  30°)]  3
Identify the values of k, a, b, and h.
π ⎤ 55. y = 2 sec ⎡⎣3 θ  _ +2 2 ⎦
k = 4, so the vertical shift is 4.
π ⎤ 1 1 θ+_ 56. y = _ cos ⎡⎣_ 1 2 4 ⎦ 4
2π _ b = 3, so the period is _ or 2π .
2
)
(
a = 2, so the amplitude is 2.
)
(
3 π π h = _ , so the phase shift is _ to the left. 2 2 ⎪3⎥
1 1 (θ  90°)⎤ + 2 57. y = _ sin ⎡⎣_ ⎦ 3
3
58. The graph below approximates the height y of a rope that two people are twirling as a function of time t in seconds. Write an equation for the function. 10
[ ( _2 )] + 4. Then graph the function.
y 6 4
y
2
8 0
6
π 2
π
3π 2
θ
4 2 0
1
2
3
t
connectED.mcgrawhill.com
881
Study Guide and Review Continued CLE 3103.4.5, ✔3103.4.6
1399 Inverse Trigonometric Functions
(pp. 871–876)
Evaluate each inverse trigonometric function. Write angle measures in degrees and radians. 59. Sin1 (1) 60. Arctan (0)
Example 12 1 Evaluate Cos1 _ . Write angle measures in degrees 2 and radians. 1 Find the angle θ for 0° ≤ θ ≤ 180° that has a cosine value of _ . 2
√ 3 2
61. Arcsin _
Use a unit circle. 1
√ 2 62. Cos1 _ 2
y
( 12 , √23 ) π
63. Tan1 1
θ 0
1
64. Arccos 0 65. RAMPS A bicycle ramp is 5 feet tall and 10 feet long, as shown below. Write an inverse trigonometric function that can be used to find θ, the angle the ramp makes with the ground. Then find the angle.
60° = 3 1 x
1
1 Find a point on the unit circle that has an xcoordinate of _ . 2
1 When θ = 60°, Cos θ = _ . 10 ft
5 ft
θ
π So, Cos1 = 60° or _ .
2
3
Example 13
_)
(
Evaluate sin Tan1 1 . Round to the nearest hundredth. 2
Evaluate each inverse trigonometric function. Round to the nearest hundredth if necessary.
Use a calculator.
1 66. tan Cos1 _
KEYSTROKES:
(
3)
(
√ 2 67. Sin Arcsin _ 2
[TAN 1] 1
2
0.4472135955
)
68. sin (Tan1 0)
1 So, sin Tan1 _ ≈ 0.45.
(
2
)
Example 14 If Cos θ = 0.72, find θ.
Solve each equation. Round to the nearest tenth if necessary.
Use a calculator.
69. Tan θ = 1.43
KEYSTROKES:
70. Sin θ = 0.8
So, θ ≈ 43.9°.
71. Cos θ = 0.41
882  Chapter 13  Study Guide and Review
[COS 1] .72
43.9455195623
Practice Test
Tennessee Curriculum Standards
Solve ABC by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
18. NAVIGATION Airplanes and ships measure distance in nautical miles. The formula 1 nautical mile = 6077  31 cos 2θ feet, where θ is the latitude in degrees, can be used to find the approximate length of a nautical mile at a certain latitude. Find the length of a nautical mile when the latitude is 120°.
"
b
$
SPI 3103.4.3
c
a
Find the amplitude and period of each function. Then graph the function.
#
19. y = 2 sin 3θ
1. A = 36°, c = 9
1 20. y = _ cos 2θ 2
2. a = 12, A = 58° 21. MULTIPLE CHOICE What is the period of the function y = 3 cot θ?
3. B = 85°, b = 8 4. a = 9, c = 12
F 120°
Rewrite each degree measure in radians and each radian measure in degrees. 5. 325°
6. 175°
9π 7. _
5π 8. _
G 180° H 360° J 1080°
6
4
9. Determine whether ABC, with A = 110°, a = 16, and b = 21, has no solution, one solution, or two solutions. Then solve the triangle, if possible. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
22. Determine whether XYZ, with y = 15, z = 9, and X = 105°, should be solved by beginning with the Law of Sines or Law of Cosines. Then solve the triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
Find the exact value of each function. Write angle measures in degrees. 10. cos (90°)
11. sin 585°
4π 12. cot _
9π 13. sec _
(
3
14. tan
(
Cos1
_4 5
)
4
State the amplitude, period, and phase shift for each function. Then graph the function.
)
23. y = cos (θ + 180)
π 1 24. y = _ tan θ  _ 2
1 15. Arccos _ 2
16. The terminal side of angle θ in standard position
(
)
√ 3 1 _ intersects the unit circle at point P _ , . 2 2
Find cos θ and sin θ. 17. MULTIPLE CHOICE What angle has a tangent and sine that are both negative?
(
2
)
25. WHEELS A water wheel has a diameter of 20 feet. It makes one complete revolution in 45 seconds. Let the height at the top of the wheel represent the height at time 0. Write an equation for the height of point h in the diagram below as a function of time t. Then graph the function. h
A 65° B 310° C 120° D 265° connectED.mcgrawhill.com
883
Preparing for Standardized Tests Using a Scientific Calculator Scientific calculators and graphing calculators are powerful problemsolving tools. As you have likely seen, some test problems that you encounter have steps or computations that require the use of a scientific calculator.
Strategies for Using a Scientific Calculator Step 1 Familiarize yourself with the various functions of a scientific calculator as well as when they should be used. • Scientific notation—for calculating large numbers • Logarithmic and exponential functions—growth and decay problems, compound interest • Trigonometric functions—problems involving angles, triangle problems, indirect measurement problems • Square roots and nth roots—distance on a coordinate plane, Pythagorean Theorem
Step 2 Use your scientific or graphing calculator to solve the problem. • Remember to work as efficiently as possible. Some steps may be done mentally or by hand, while others must be done using your calculator. • If time permits, check your answer.
SPI 3108.4.15
Test Practice Example Read the problem. Identify what you need to know. Then use the information in the problem to solve. When Molly stands at a distance of 18 feet from the base of a tree, she forms an angle of 57° with the top of the tree. What is the height of the tree to the nearest tenth? A 27.7 ft B 28.5 ft C 29.2 ft D 30.1 ft
884  Chapter 13  Preparing for Standardized Tests
Read the problem carefully. You are given some measurements and asked to find the height of a tree. It may be helpful to first sketch a model of the problem.
h
Molly
57° 18 ft
Use a trigonometric function to relate the lengths and the angle measure in the right triangle. opposite adjacent h tan 57° = _ 18
tangent θ = _
Definition of tangent ratio Substitute.
You need to evaluate tan 57º to solve for the height of a tree h. Use a scientific calculator h 1.53986 ≈ _ 18
27.71748 ≈ h
Use a calculator. Multiply each side by 18.
The height of the tree is about 27.7 feet. The correct answer is A.
Exercises Read each problem. Identify what you need to know. Then use the information in the problem to solve.
2. What is the angle of the bike ramp below?
1. An airplane takes off and climbs at a constant rate. After traveling 800 yards horizontally, the plane has climbed 285 yards vertically. What is the plane’s angle of elevation during the takeoff and initial climb?
12 ft
θ 10 ft
A 15.6°
F 26.3°
B 18.4°
G 28.5°
C 19.6°
H 30.4°
D 22.3°
J
33.6° connectED.mcgrawhill.com
885
Standardized Test Practice Cumulative, Chapters 1 through 13 6. What is the solution of the system of equations shown below?
Multiple Choice Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1. What is the value of x? Round to the nearest tenth if necessary.
64° x
A 6.5
15
B 6.9
xy+z=0 5x + 3y  2z = 1 2x  y + 4z = 11 F (0, 3, 3)
H no solution
G (2, 5, 3)
J
infinitely many solutions
7. Find m in triangle MNO if n = 12.4 centimeters, M = 35°, and N = 74°. Round to the nearest tenth.
C 7.1
A 7.4 cm
C 14.6 cm
D 7.3
B 8.5 cm
D 35.9 cm
2. Marvin rides his bike at a speed of 21 miles per hour and can ride his training loop 10 times in the time that it takes his younger brother to complete the training loop 8 times. Which is a reasonable estimate for Marvin’s younger brother’s speed?
8. The results of a recent poll are organized in the matrix. For Against Proposition 1 ⎡ 1553 Proposition 2 ⎢ 689 Proposition 3 ⎣ 2088
F between 14 mph and 15 mph G between 15 mph and 16 mph H between 16 mph and 17 mph J
between 17 mph and 18 mph
3. Suppose a Ferris wheel has a diameter of 68 feet. The wheel rotates 12° each time a new passenger is picked up. How far would you travel when the wheel rotates 12°? Round to the nearest tenth if necessary. A 7.1 ft
C 7.8 ft
B 7.5 ft
D 14.2 ft
4. What is the slope of a line parallel to y  2 = 4(x + 1)? F 4
1 H _
1 G _
J
4
771 ⎤ 1633 229 ⎦
Based on these results, which conclusion is NOT valid? F There were 771 votes cast against Proposition 1. G More people voted against Proposition 1 than voted for Proposition 2. H Proposition 2 has little chance of passing. J
More people voted for Proposition 1 than for Proposition 3.
9. The graph of which of the following equations is symmetrical about the yaxis? A y = x 2 + 3x  1
C y = 6x 2 + 9
B y = x 2 + x
D y = 3x 2  3x + 1
10. What is the remainder when x3  7x + 5 is divided by x + 3?
4
4
F 11
G 1
H 1
J
5. What is the exact value of sin 240º? √ 3 2
1 A _
C _
√2 B _
√ 3 D _
2
3
2
886  Chapter 13  Standardized Test Practice
TestTakingTip Question 7 Use the Law of Sines to solve the triangle.
11
16. GRIDDED RESPONSE The pattern of squares below continues infinitely, with more squares being added at each step. How many squares are in the tenth step?
Short Response/Gridded Response Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 11. The speed a tsunami, or tidal wave, can travel is modeled by the equation s = 356 √ d , where s is the speed in kilometers per hour and d is the average depth of the water in kilometers. A tsunami is found to be traveling at 145 kilometers per hour. What is the average depth of the water? Round to the nearest hundredth.
4UFQ
12. GRIDDED RESPONSE Suppose you deposit $500 in an account paying 4.5% interest compounded semiannually. Find the dollar value of the account rounded to the nearest penny after 10 years.
4UFQ
Extended Response
13. In order to remain healthy, a horse requires 10 pounds of hay per day.
Record your answers on a sheet of paper. Show your work.
a. Write an equation to represent the amount of hay needed to sustain x horses for d days. b. Is your equation a direct, joint, or inverse variation? Explain.
17. Amanda’s hours at her summer job for one week are listed in the table below. She earns $6 per hour. Amanda’s Work Hours Sunday 0 Monday 6 Tuesday 4 Wednesday 0 Thursday 2 Friday 6 Saturday 8
c. How much hay do three horses need for the month of July? 14. GRIDDED RESPONSE What is the radius of the circle with equation x 2 + y 2 + 8x + 8y + 28 = 0? 15. Anna is training to run a 10kilometer race. The table below lists the times she received in several 1kilometer races. The times are listed in minutes. What was her mean time in minutes for a 1kilometer race? 7.25 7.40 7.20 7.10 8.00
4UFQ
a. Write an expression for Amanda’s total weekly earnings. b. Evaluate the expression from part a by using the Distributive Property.
8.10 6.75 7.35 7.25 7.45
c. Michael works with Amanda and also earns $6 per hour. If Michael’s earnings were $192 this week, write and solve an equation to find how many more hours Michael worked than Amanda.
Need ExtraHelp? If you missed Question... Go to Lesson... For help with TN SPI...
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
131
13
132
24
133
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134
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103 122 112
13
3108. 4.15
3103. 3.13
3108. 4.8
3102. 1.6
3103. 4.1
3103. 3.8
3.7
3102. 5.1
3103. 3.11
3103. 3.1
3103. 3.13
3103. 3.13
3102. 1.2
3103. 3.11
3102. 1.3
3103. 5.1
16
3103. 3.4
connectED.mcgrawhill.com
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887