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 b c

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.

hsm11gmse_NA_0803.indd 507 4/15/09 11:36:13 AM

http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html

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 x150 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.

hsm11gmse_NA_0803.indd 508 2/25/09 11:56:26 PM

http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html

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 5 610 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

T100

41

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

Use the cosine ratio.

cos X 5 1520

m/X 5 cos21Q 15 20R

cos–1 15 20 enter

m/X < 41.40962211

< 41

hsm11gmse_NA_0803.indd 509 2/25/09 11:56:32 PM

http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html

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

hsm11gmse_NA_0803.indd 510 2/25/09 11:56:37 PM

http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html

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.