Sine and Cosine Ratios GEOMETRY LESSON 9-2

(For help, go to Lesson 9-1.)

For each triangle, find (a) the length of the leg opposite length of the leg adjacent to B. 1.

2.

3.

B and (b) the

Sine and Cosine Ratios GEOMETRY LESSON 9-2

Solutions 1. a. The leg opposite B is the one that is not a side of the angle: 9. b. The leg adjacent to B is the one that is a side of the angle: 12.

2. a. The leg opposite B is the one that is not a side of the angle: 7. b. The leg adjacent to B is the one that is a side of the angle: 2 78. 3. a. The leg opposite B is the one that is not a side of the angle: 10. b. The leg adjacent to B is the one that is a side of the angle: 3 29.

Sine and Cosine Ratios GEOMETRY LESSON 9-2

Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms. 12 3 opposite = = sin T = hypotenuse 20 5 16 4 adjacent = = cos T = hypotenuse 20 5 4 16 opposite = = sin G = hypotenuse 20 5 12 3 adjacent = = cos G = hypotenuse 20 5

Sine and Cosine Ratios GEOMETRY LESSON 9-2

A 20-ft. wire supporting a flagpole forms a 35˚ angle with the flagpole. To the nearest foot, how high is the flagpole? The flagpole, wire, and ground form a right triangle with the wire as the hypotenuse. Because you know an angle and the measures of its adjacent side and the hypotenuse, you can use the cosine ratio to find the height of the flagpole. height cos 35° = 20

height = 20 • cos 35°

20

35

16.383041

Use the cosine ratio. Solve for height.

Use a calculator.

The flagpole is about 16 ft tall.

Sine and Cosine Ratios GEOMETRY LESSON 9-2

A right triangle has a leg 1.5 units long and hypotenuse 4.0 units long. Find the measures of its acute angles to the nearest degree. Draw a diagram using the information given. Use the inverse of the cosine function to find m A. 1.5

cos A = 4.0 = 0.375

Use the cosine ratio.

m A = cosâ€“1(0.375)

Use the inverse of the cosine.

0.375

67.975687

m A

68

Use a calculator. Round to the nearest degree.

Sine and Cosine Ratios GEOMETRY LESSON 9-2

(continued)

To find m B, use the fact that the acute angles of a right triangle are complementary. m A + m B = 90 68 + m B

90

m B

22

Definition of complementary angles Substitute.

The acute angles, rounded to the nearest degree, measure 68 and 22.

Sine and Cosine Ratios GEOMETRY LESSON 9-2

Pages 479-481 Exercises 1.

7 24 ; 25 25

2.

4

3. 4.

9. 106.5

17. about 17 ft 8 in.

2 ; 7 9 9 1 ; 3 2 2

10. 1085 ft

18. sin X ÷ cos X =

11.5

12. 51

11. 21

opp. adj. opp. ÷ = hyp. hyp. adj.

= tan X 19. cos X • tan X =

5.

8.3

13. 46

6.

17.9

14. 59

7.

17.0

15. 24

8.

4.3

16. 66

opp. adj. opp. • = adj. hyp. hyp.

= sin X 20. sin X ÷ tan X = opp. opp. adj. ÷ = hyp. adj. hyp.

= cos X

Sine and Cosine Ratios GEOMETRY LESSON 9-2

21. No; the s are ~ and the sine ratio for 35째 is constant. 22. w = 3; x = 41.4 23. w = 37; x = 7.5 24. w = 68.3; x = 151.6 25. a. They are equal; yes; the sine and cosine of complementary s are =.

b.

B;

A

c. Answers may vary. Sample: cosine of A = sine of the compl. of A. 26. a.

2 2 2

b. 2

c.

2 2

d.

2 2

e. They are equal. 27. Yes; use any trig. function and the known measures to find one other side. Use the Pythagorean Thm. to find the 3rd side. Subtract the acute measure from 90 to get the other measure.

Sine and Cosine Ratios GEOMETRY LESSON 9-2

28. a.

b. c.

3

29. (continued) opp. Since sin A = ,

2 1 2 1 2

hyp.

if sin A ≥ 1, then opp. ≥ hyp., which is impossible. 3

d.

d. For s that approach 90, the opp. side gets close to the hyp. in length, so opp. hyp.

approaches 1.

2

30. a. 0.99985

e. cos 30° = 30°

3 sin

f. sin 60° = 60°

3 cos

31. (sin A)2 + (cos A)2 = b-d. Answers may vary, samples are given. b. 1.

29. Answers may vary. Sample:

c. sin X = 1 for X = 89.9; no

a 2+ c

b 2= c

a2 + b2 = c2 c2 c2 a2 + b2 = c2 = 1 c2

Sine and Cosine Ratios GEOMETRY LESSON 9-2

32. (sin B)2 + (cos B)2 = 2 b 2 + a = c c a2 b2 + c2 = c2

c2 b2 + a2 = 2 =1 c c2

33.

1 – (tan A)2 = (cos A)2 2 1 ÷ b2

c

2 – a2 =

34.

1 1 – = 2 2 (sin A) (tan A)

c2 – b2 = a2 a2

1 = a 2 c

a2 c 2 – b2 = =1 a2 a2

2 2 2 2 35. (tan A)2 – (sin A)2 = a – a = a2 – a2 =

b

a2c2 – a2b2 = a2c2 – a2b2 b2c2 b2c2 b2c2

c

=

c b a2(c2 – b2) = b2c2

a2 • a2 = a 2 a 2 = (tan A)2(sin A)2 c b2 • c2 b

b

c2 – a2 = c2 + a2 = b2 b2 b2 b2 = 1 b2

1 – a 2 b

36. a. about 1.5 AU b. about 5.2 AU

Sine and Cosine Ratios GEOMETRY LESSON 9-2

37. A

38. H

40. (continued) [1] one angle found correctly

39. A

41. 6.9

40. [2] a. cos G = 7

42. 3.3

10

m G = cos–1 ≈ 46

7 10

b. m R ≈ 90 – 46 = 44 OR m R = sin–1 7

10

43. 18 44. 12.9 ft or 4.9 ft 45. (36 + 18

3) cm2

46. 42.5 in2 ≈ 44

47. 30

3 mm2

Sine and Cosine Ratios GEOMETRY LESSON 9-2

Use this figure for Exercises 1 and 2. 1. Write the ratios for sin A and sin B. 8 16 sin A = or 17 , sin B = 30 or 15 34

34

17

2. Write the ratios for cos A and cos B. 8 30 15 16 cos A = or , cos B = or 34

17

34

Use this figure for Exercises 3 and 4. 3. Find x to the nearest tenth. 21.0 4. Find y to the nearest tenth. 13.6 Use this figure for Exercises 5 and 6. 5. Find x to the nearest degree. 44 6. Find y to the nearest degree. 46

17

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