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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.
Your Digital Math Portal Personal Tutor
Tennessee Curriculum Standards CLE 3103.4.5
Get Ready for the Chapter Diagnose Readiness
You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
Find the value of x to the nearest tenth. (Lesson 0-7) 1.
Example 1 Find the missing measure of the right triangle.
c2 = a2 + b2
18 = a + 5
324 = a 2 + 25
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)
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.
x + x = 18
2x 2 = 18 2
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?
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
Online Option Take an online self-check Chapter Readiness Quiz at connectED.mcgraw-hill.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.
Trigonometric Functions Make this Foldable to help you organize your Chapter 13 notes about trigonometric functions. Begin with four pieces of grid paper.
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.
angle of elevation
ángulo de depresión
angle of depression
ángulo de elevación
Law of Sines
Ley de los senos
Law of Cosines
Ley de los cosenos
p. 855 p. 856
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
Investigating Special Right Triangles You can use a spreadsheet to investigate side measures of special right triangles.
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)
✔ 3103.5.2 Organize and display data using appropriate methods (including spreadsheets and technology tools) to detect patterns and departures from patterns.
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
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.mcgraw-hill.com
Trigonometric Functions in Right Triangles Then
You used the Pythagorean Theorem to find side lengths of right triangles.
Find values of trigonometric functions for acute angles.
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
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.
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.
sin (sine) θ = _
Tennessee Curriculum Standards CLE 3103.4.5 Use trigonometric concepts, properties and graphs to solve problems.
opp hyp adj cos (cosine) θ = _ hyp opp tan (tangent) θ = _ adj
_ 5 5 csc θ = _ sin θ = 4
csc (cosecant) θ = _ opp hyp adj adj cot (cotangent) θ = _ opp
sec (secant) θ = _
_ 5 _ sec θ = 5 cos θ = 3 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
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 13-1
StudyTip Memorize Trigonometric Ratios SOH-CAH-TOA 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 = _
1 cot 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.
Step 2 Use the Pythagorean Theorem to find a. a2 + b2 = c2
b = 5 and c = 8
a2 + 25 = 64 a2
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
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.
adj √ 39 _ cot B = _ opp = 5
opp 5 √ 39 5 tan B = _ = _ or _ 39 adj √ 39
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° = _
tan 60° = √ 3 x √⎯ 3
45°-45°-90° √ 2 2
sin 45° = _
√ 2 2
cos 45° = _
tan 45° = 1
x √⎯ 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.
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
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.
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?
Real-WorldLink 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
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
Replace θ with 76°, opp with d, and adj with 200. Multiply each side by 200. Use a calculator to simplify: 200
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 13-1 | Trigonometric Functions in Right Triangles
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 sin-1 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 sin-1 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.
If sin A = x, then sin-1 x = m∠A.
1 1 sin A = _ → sin-1 _ = m∠A → m∠A = 30°
If ∠A is an acute angle and the cosine of A is x, then the inverse cosine of x is the measure of ∠A.
If cos A = x, then cos-1 x = m∠A.
cos A = _ → cos-1 _ = m∠A → m∠A = 45°
If ∠A is an acute angle and the tangent of A is x, then the inverse tangent of x is the measure of ∠A.
If tan A = x, then tan-1 x = m∠A.
tan A = √ 3 → tan-1 √ 3 = m∠A → m∠A = 60°
this notation is similar to the notation for an inverse function, f-1(x).
√ 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 .
You know the measure of the side opposite ∠N and the measure of the hypotenuse. Use the sine function. 6 sin N = _
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 cos-1 _ = m∠B
60° = m∠B
cos θ =
adj _ hyp
Inverse cosine Use a calculator.
GuidedPractice Find x. Round to the nearest tenth if necessary. 5A.
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.
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.
Divide each side by sin 12°.
x ≈ 173.2
Use a calculator.
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.
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).
Real-WorldLink 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.
Divide each side by sin 60°.
x ≈ 225.2
Use a calculator.
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 14-ft 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 13-1 | Trigonometric Functions in Right Triangles
Check Your Understanding Example 1
Find the values of the six trigonometric functions for angle θ. 1.
= Step-by-Step Solutions begin on page R20.
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?
Examples 3–4 Use a trigonometric function to find the value of x. Round to the nearest tenth. 5
Find the value of x. Round to the nearest tenth. 15
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 21-foot 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
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
Examples 3–4 Use a trigonometric function to find each value of x. Round to the nearest tenth. 9
x 30° 18
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.
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.
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.
Use trigonometric functions to find the values of x and y. Round to the nearest tenth. 37.
50 ° 30.2 46.5°
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
814 | Lesson 13-1 | 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.
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.
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° "
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
Use Higher-Order 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 x-axis? 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.
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.
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 ⎦
⎡ 2 ⎣ -3
-1 ⎤ 2⎦
b. What is the least amount that you would have to spend for the yearbooks?
59. SHORT RESPONSE As a fundraiser, the marching band sold T-shirts 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 T-shirt was $15, how many T-shirts were sold?
⎡ 1 ⎣ -1
⎡ 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
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 12-7) 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 12-6) 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
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
Find each probability. (Lesson 12-3) 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 6-1)
5280 feet 69. 4.3 miles _
18 cubic inches 72. __ 24 seconds 5 seconds
8 pints 70. 8 gallons _ 1 gallon
10 centimeters 73. 65 degrees __
816 | Lesson 13-1 | Trigonometric Functions in Right Triangles
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.
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.
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 x-axis. • The ray on the x-axis 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
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.
If the measure of an angle is positive, the terminal side is rotated counterclockwise.
If the measure of an angle is negative, the terminal side is rotated clockwise.
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 x-axis. y
The angle is negative. Draw the terminal side of the angle 40° clockwise from the positive x-axis. y
GuidedPractice 1A. 80°
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
Real-World Example 2 Draw an Angle in Standard Position WAKEBOARDING Wakeboarding is a combination of surfing, skateboarding, snowboarding, and water skiing. One maneuver involves a 540-degree 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 x-axis.
Wakeboarding is one of the fastest-growing 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°.
• 60° + 360° = 420° • 60° - 360° = -300°
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°
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.
Convert Between Degrees and Radians Angles can
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.
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°
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
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.
π radians -30° = -30° · _ 180°
5π 5π 180° _ =_ radians · _ 2
-30π π =_ or -_ radians 180
900° =_ or 450° 2
GuidedPractice 3π 4B. -_
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
120° 135° 150°
7π 6 5π 4
60° 45° 30°
0 2π x ° 360
330° 315° 300°
210° 225° 240° 4π 3
270° 3π 2
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
For a circle with radius r and central angle θ (in radians), the arc length s equals the product of r and θ.
s = rθ You will justify this formula in Exercise 52
Real-World 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.
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.
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
= Step-by-Step Solutions begin on page R20.
Examples 1–2 Draw an angle with the given measure in standard position. 1. 140° Example 3
Rewrite each degree measure in radians and each radian measure in degrees. π 7. _ 4
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. 4. 25°
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 13-2 | 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°
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
Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle. 19. 50°
Rewrite each degree measure in radians and each radian measure in degrees. 25 330°
5π 26. _
π 27. -_
7π 30. -_
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
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°
3π 39. -_ 4
19π 40. _ 6
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).
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. _
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.
a. Find the angle θ in radians through which the carousel rotates in one second.
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?
Use Higher-Order 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)°.
The measure of a coterminal angle is (360 – x)°.
π 49. CHALLENGE A line makes an angle of _ radians with the positive x-axis 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 13-2 | 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
F 2 √ 34
H 4 √ 109
G 2 √ 109
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 13-1) 58.
A binomial distribution has a 40% rate of success. There are 12 trials. (Lesson 12-7) 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 12-5) 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 11-5)
Skills Review Use the Pythagorean Theorem to find the length of the hypotenuse for each right triangle with the given side lengths. (Lesson 0-7) 65. a = 12, b = 15
66. a = 8, b = 17
67. a = 14, b = 11 connectED.mcgraw-hill.com
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
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
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
30° 9 in.
824 | Extend 13-2 | Geometry Lab: Areas of Parallelograms
100 ft 75° 200 ft
Trigonometric Functions of General Angles Then
You found values of trigonometric functions for acute angles. (Lesson 13-1)
Find values of trigonometric functions for general angles.
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
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.
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
1 (x, y )
sec θ = _r , x ≠ 0 cot θ = _x , y ≠ 0 x
tan θ = _, x ≠ 0
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.
Step 1 Draw the angle, and find the value of r. r=
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.
If the terminal side of angle θ in standard position lies on the x- or y-axis, 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 (0, r ) (r, 0) x
θ = 270° 3π or _ radians 2
θ = 180° or π radians
(-r, 0) 0
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.
y 6 r r _ _ csc θ = = 6 or 1 6 y
The point at (0, 6) lies on the positive y-axis, 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
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.
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 xx-axis. The rules for finding the measures of reference angles for 0° < θ < 360° or 0° < θ < 2π are shown below. fo
KeyConcept Reference Angles 2VBESBOU*
θ = θ
θ = 180° - θ θ = π - θ
826 | Lesson 13-3 | Trigonometric Functions of General Angles
θ = θ - 180° θ = θ - π
θ = 360° - θ θ = 2π - θ
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 13-2 Concept Summary to help you sketch angles.
Sketch each angle. Then find its reference angle.
b. - 5π
3π 5π coterminal angle: -_ + 2π = _
θ = 210° θ'
θ = 3π 4
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
GuidedPractice 2π 3B. _
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.
sin θ, csc θ: +
sin θ, csc θ: +
cos θ, sec θ: -
cos θ, sec θ: +
tan θ, cot θ: -
tan θ, cot θ: +
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 13-1.
Sine 1 sin 30° = _ 2 √2 _ sin 45° = 2 √3 _ sin 60° = 2
Trigonometric Values for Special Angles Tangent Cosecant
√ 3 cos 30° = _ 2 √ 2 cos 45° = _ 2
√ 3 tan 30° = _ 3
1 cos 60° = _ 2
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
√ 3 3
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.
θ = π - θ 6
5π π csc _ = csc _ 6
θ = 5π
Find the measure of the reference angle.
5π π =π-_ or _ 6
= csc 30° =2
θ' = π 6
The cosecant function is positive in Quadrant II.
_π radians = 30° 6 1 csc 30° = _ sin 30
GuidedPractice 5π 4B. tan _
4A. cos 135°
Real-World 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°
reference angle: 180° - 160° = 20° y r y _ sin 20° = 84
sin θ = _
Real-WorldLink 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 13-3 | Trigonometric Functions of General Angles
Check Your Understanding
= Step-by-Step 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°
3π 6. -_
Find the exact value of each trigonometric function. 3π 7. sin _ 4
3. (0, -4)
5π 8. tan _
9. sec 120°
11. ENTERTAINMENT Alejandra opens her portable DVD player so that it forms a
10. sin 300°
5 2 in.
1 125° angle. The screen is 5_ inches long.
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.
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°
7π 21. _
π 22. -_
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 _
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.
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.
b. About how far away from the goalie was the soccer player? connectED.mcgraw-hill.com
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?
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?
68 ft x
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
Find the exact value of each trigonometric function. 41. cot 270° 7π 44. tan -_
42. csc 180°
43. sin 570°
11π 45. cos -_
9π 46. cot _ 4
Use Higher-Order 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 13-3 | Trigonometric Functions of General Angles
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
1 E _ -1 4
Spiral Review Rewrite each radian measure in degrees. (Lesson 13-2) 4 56. _ π
11 57. _ π
17 58. -_ π
Solve each equation. (Lesson 13-1) 13 59. cos a = _
b 60. sin 30 = _
9 61. tan c = _
62. ARCHITECTURE A memorial being constructed in a city park will be a brick wall, with a top row of six gold-plated 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 11-7) 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 30-day 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 11-3) Write an equation for each circle given the endpoints of a diameter. (Lesson 10-3) 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 9-2) 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 ten-thousandth. (Lesson 8-6) 70. 8x = 30
72. 3x + 2 = 41
71. 5x = 64
Evaluate each expression. (Lesson 7-6) 73. 16
74. 27 3
5 -_ 2
Skills Review Solve for x. (Concepts and Skills Bank 1) x+2 x-2 76. _ = _ 18
x+5 7 77. _ = _ x-1
15 5 78. _ =_ x+8
2x + 20
Law of Sines Then
You found side lengths and angle measures of right triangles.
Find the area of a triangle using two sides and an included angle.
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.
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.
Find the Area of a Triangle In the triangle at h the right, sin A = _ , or h = c sin A.
1 Area = _ bh
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.
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.
1 1 1 Area = _ bc sin A = _ ac sin B = _ ab sin C 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
Based on the known measures, use the third area formula.
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
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 13-4
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
Set the area formulas equal to each other.
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
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.
sin A sin B sin C _ =_ =_ b a c
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 (angle-angle-side AAS or angle-side-angle ASA cases)
• the measures of two sides and the angle opposite one of the sides (side-side-angle SSA case)
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
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°
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
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.
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.