Carlos plans to build a grain bin

Lesson 8-3 Trigonometry 507

8-3 Trigonometry

Objective To use the sine, cosine, and tangent ratios to determine side lengths and angle measures in right triangles

What is the ratio of the length of the shorter leg to the length of the hypotenuse for each of kADF, kAEG, and kABC? Make a conjecture based on your results.

Essential Understanding If you know certain combinations of side lengths and angle measures of a right triangle, you can use ratios to fi nd other side lengths and angle measures.

Any two right triangles that have a pair of congruent acute angles are similar by the AA Similarity Postulate. Similar right triangles have equivalent ratios for their corresponding sides called trigonometric ratios.

B

E

D

A F G C62 4

4

Dynamic Activity Trigonometric Ratios T

A C

T I V I T I

E S

D TT

AAAAAAAA C

A CC

I E SSSSSSSS

DY NAMIC

Lesson Vocabulary

• trigonometric ratios

• sine • cosine • tangent

L V L V

• t

LL VVV

• t

Key Concept Trigonometric Ratios

sine of /A 5 length of leg opposite /A

length of hypotenuse 5

a c

cosine of /A 5 length of leg adjacent to /A

length of hypotenuse 5 bc

tangent of /A 5 length of leg opposite /A

length of leg adjacent to /A 5

a b

A

B

C

c a

b

Here are ratios in triangles once again! This must be “similar” to something you’ve seen before.

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5º

150 ft

Problem 1

Got It?

Problem 2

508 Chapter 8 Right Triangles and Trigonometry

You can abbreviate the ratios as

sin A 5 opposite

hypotenuse , cos A 5

adjacent hypotenuse

, and tan A 5 opposite adjacent

.

Writing Trigonometric Ratios

What are the sine, cosine, and tangent ratios for lT ?

sin T 5 opposite

hypotenuse 5

8 17

cos T 5 adjacent

hypotenuse 5

15 17

tan T 5 opposite adjacent

5 8

15

1. Use the triangle in Problem 1. What are the sine, cosine, and tangent ratios for /G?

In Chapter 7, you used similar triangles to measure distances indirectly. You can also use trigonometry for indirect measurement.

Using a Trigonometric Ratio to Find Distance

Landmarks In 1990, the Leaning Tower of Pisa was closed to the public due to safety concerns. Th e tower reopened in 2001 after a 10-year project to reduce its tilt from vertical. Engineers’ eff orts were successful and resulted in a tilt of 58, reduced from 5.58. Suppose someone drops an object from the tower at a height of 150 ft. How far from the base of the tower will the object land? Round to the nearest foot.

Th e given side is adjacent to the given angle. Th e side you want to fi nd is opposite the given angle.

tan 58 5 x

150 Use the tangent ratio.

x 5 150(tan 58) Multiply each side by 150.

150 tan 5 enter Use a calculator.

x < 13.12329953

Th e object will land about 13 ft from the base of the tower.

17 8

G

RT 15

G

W

G

How do the sides relate to lT ? GR is across from, or opposite, /T . TR is next to, or adjacent to, /T . TG is the hypotenuse because it is opposite the 908 angle.

n

Th s

What is the fi rst step? Look at the triangle and determine how the sides of the triangle relate to the given angle.

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Got It?

Problem 3

Got It?

Lesson 8-3 Trigonometry 509

G

When should you use an inverse? Use an inverse when you know two side lengths of a right triangle and you want to fi nd the measure of one of the acute angles.

2. For parts (a)–(c), fi nd the value of w to the nearest tenth. a. b. c.

d. A section of Filbert Street in San Francisco rises at an angle of about 178. If you walk 150 ft up this section, what is your vertical rise? Round to the nearest foot.

If you know the sine, cosine, or tangent ratio for an angle, you can use an inverse

(sin21, cos21, or tan21) to fi nd the measure of the angle.

Using Inverses

What is mlX to the nearest degree?

A B

You know the lengths of the hypotenuse and the side opposite /X .

Use the sine ratio.

sin X 5 6

10 Write the ratio.

m/X 5 sin21 Q 6

10R Use the inverse.

sin–1 6 10 enter Use a calculator.

m/X < 36.86989765

< 37

3. a. Use the fi gure at the right. What is m/Y to the nearest degree?

b. Reasoning Suppose you know the lengths of all three sides of a right triangle. Does it matter which trigonometric ratio you use to fi nd the measure of any of the three angles? Explain.

w17

54 w

28

1.0

33 4.5

w

H

B

6 10

X

15

M

X

N

20

P

Y

T 100

41

You know the lengths of the hypotenuse and the side adjacent to /X .

Use the cosine ratio.

cos X 5 15 20

m/X 5 cos21Q 15 20R

cos–1 15 20 enter

m/X < 41.40962211

< 41

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Lesson Check

510 Chapter 8 Right Triangles and Trigonometry

Practice and Problem-Solving Exercises

Write the ratios for sin M, cos M, and tan M.

11. 12. 13.

Find the value of x. Round to the nearest tenth.

14. 15. 16.

17. 18. 19.

20. Recreation A skateboarding ramp is 12 in. high and rises at an angle of 178. How long is the base of the ramp? Round to the nearest inch.

21. Public Transportation An escalator in the subway station has a vertical rise of 195 ft 9.5 in., and rises at an angle of 10.48. How long is the escalator? Round to the nearest foot.

PracticeA See Problem 1.

7

25

24 L

K

M

4 V2

7M K

L

9

K L

M

4 2

2 V3

See Problem 2.

35

20 x

41 11

x

64 7 x

x

36

10 62 28

50x x 10

25

Do you know HOW? Write each ratio.

1. sin A 2. cos A

3. tan A 4. sin B

5. cos B 6. tan B

What is the value of x? Round to the nearest tenth.

7. 8.

Do you UNDERSTAND? 9. Vocabulary Some people use SOH-CAH-TOA to

remember the trigonometric ratios for sine, cosine, and tangent. Why do you think that word might help? (Hint: Th ink of the fi rst letters of the ratios.)

10. Error Analysis A student states that sin A . sin X because the lengths of the sides of nABC are greater than the lengths of the sides of nXYZ . What is the student’s error? Explain.

35 C

35

Y

Z X

B

A

A

C B

10

8

6

39 15

x 27

32x

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Lesson 8-3 Trigonometry 511

Find the value of x. Round to the nearest degree.

22. 23. 24.

25. 26. 27.

28. Th e lengths of the diagonals of a rhombus are 2 in. and 5 in. Find the measures of the angles of the rhombus to the nearest degree.

29. Think About a Plan Carlos plans to build a grain bin with a radius of 15 ft. Th e recommended slant of the roof is 258. He wants the roof to overhang the edge of the bin by 1 ft. What should the length x be? Give your answer in feet and inches.

• What is the position of the side of length x in relation to the given angle?

• What information do you need to fi nd a side length of a right triangle?

• Which trigonometric ratio could you use?

An identity is an equation that is true for all the allowed values of the variable. Use what you know about trigonometric ratios to show that each equation is an identity.

30. tan X 5 sin Xcos X 31. sin X 5 cos X ? tan X 32. cos X 5 sin X tan X

Find the values of w and then x. Round lengths to the nearest tenth and angle measures to the nearest degree.

33. 34. 35.

36. Pyramids All but two of the pyramids built by the ancient Egyptians have faces inclined at 528 angles. Suppose an archaeologist discovers the ruins of a pyramid. Most of the pyramid has eroded, but the archaeologist is able to determine that the length of a side of the square base is 82 m. How tall was the pyramid, assuming its faces were inclined at 528? Round your answer to the nearest meter.

See Problem 3.

5 x

14

5

x 8

x

9

13

3.0 5.8x 17

41

x 0.15

0.34

x

ApplyB

1 ft over- hang

x

15 ft

25

6 4x w

30 10

xw 56 34

102 102

x

w

42

52

82 m82 m

52

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512 Chapter 8 Right Triangles and Trigonometry

37. a. In nABC at the right, how does sin A compare to cos B? Is this true for the acute angles of other right triangles?

b. Reading Math Th e word cosine is derived from the words complement’s sine. Which angle in nABC is the complement of /A? Of /B?

c. Explain why the derivation of the word cosine makes sense.

38. For right nABC with right /C , prove each of the following. a. sin A , 1 b. cos A , 1

39. a. Writing Explain why tan 608 5 !3. Include a diagram with your explanation. b. Make a Conjecture How are the sine and cosine of a 608 angle related? Explain.

Th e sine, cosine, and tangent ratios each have a reciprocal ratio. Th e reciprocal ratios are cosecant (csc), secant (sec), and cotangent (cot). Use kABC and the defi nitions below to write each ratio.

csc X 5 1sin X sec X 5 1

cos X cot X 5 1

tan X

40. csc A 41. sec A 42. cot A

43. csc B 44. sec B 45. cot B

46. Graphing Calculator Use the table feature of your graphing calculator to study sin X as X gets close to (but not equal to) 90. In the y= screen, enter Y1 5 sin X .

a. Use the tblset feature so that X starts at 80 and changes by 1. Access the table . From the table, what is sin X for X 5 89?

b. Perform a “numerical zoom-in.” Use the tblset feature, so that X starts with 89 and changes by 0.1. What is sin X for X 5 89.9?

c. Continue to zoom-in numerically on values close to 90. What is the greatest value you can get for sin X on your calculator? How close is X to 90? Does your result contradict what you are asked to prove in Exercise 38a?

d. Use right triangles to explain the behavior of sin X found above.

47. a. Reasoning Does tan A 1 tan B 5 tan (A 1 B) when A 1 B , 90? Explain. b. Does tan A 2 tan B 5 tan (A 2 B) when A 2 B . 0? Use part (a) and indirect

reasoning to explain.

Verify that each equation is an identity by showing that each expression on the left simplifi es to 1.

48. (sin A)2 1 (cos A)2 5 1 49. (sin B)2 1 (cos B)2 5 1

50. 1 (cos A) 2

2 (tan A)2 5 1 51. 1 (sin A) 2

2 1

(tan A) 2 5 1

52. Show that (tan A)2 2 (sin A)2 5 (tan A)2 ? (sin A)2 is an identity.

A

B

C

34

30

16

Proof

A

BC

15

12

9

ChallengeC

B

C

a c

b A

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Lesson 8-3 Trigonometry 513

53. Astronomy Th e Polish astronomer Nicolaus Copernicus devised a method for determining the sizes of the orbits of planets farther from the sun than Earth. His method involved noting the number of days between the times that a planet was in the positions labeled A and B in the diagram. Using this time and the number of days in each planet’s year, he calculated c and d.

a. For Mars, c 5 55.2 and d 5 103.8. How far is Mars from the sun in astronomical units (AU)? One astronomical unit is defi ned as the average distance from Earth to the center of the sun, about 93 million miles.

b. For Jupiter, c 5 21.9 and d 5 100.8. How far is Jupiter from the sun in astronomical units?

d˚ c̊

Sun

Not to scale

AA

B

B

1 AU1 AU

Ea rth’s orbit

Ou ter p

lanet’s orbit

Mixed Review 57. Th e length of the hypotenuse of a 308-608-908 triangle is 8. What are the

lengths of the legs?

58. A diagonal of a square is 10 units. Find the length of a side of the square. Express your answer in simplest radical form.

Get Ready! To prepare for Lesson 8-4, do Exercises 59–62.

Use rectangle ABCD to complete each statement.

59. /1 > 9

60. /5 > 9

61. /3 > 9

62. m/1 1 m/5 5 9

See Lesson 8-2.

See Lessons 3-2 and 6-4.

C

A B

D

10

11 1 6 8

573

Standardized Test Prep

54. Grove Street has a grade of 20%. Th at means that the street rises 20 ft for every 100 ft of horizontal distance. To the nearest tenth, at what angle does Grove Street rise?

11.38 78.58

11.58 78.78

55. Which of the following fi gures is NOT a parallelogram?

square trapezoid rhombus rectangle

56. In nABC, AB . BC . AC . One angle has a measure of 168. What are all the possible whole-number values for the measure of /A? Explain.

20 ft

100 ft

Grove StreetSAT/ACT

Short Response

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