Csg ch06 - Chapter 6 - all problem solutions PDF

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Chapter 6 - all problem solutions...

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C H A P T E R 6 Additional Topics in Trigonometry Section 6.1

Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . 532

Section 6.2

Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . 539

Section 6.3

Vectors in the Plane

Section 6.4

Vectors and Dot Products

Section 6.5

Trigonometric Form of a Complex Number . . . . . . . . 571

. . . . . . . . . . . . . . . . . . . . 549 . . . . . . . . . . . . . . . . . 562

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Practice Test

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

C H A P T E R 6 Additional Topics in Trigonometry Section 6.1

■

Law of Sines

If ABC is any oblique triangle with sides a, b, and c, then c b a . ⫽ ⫽ sin A sin B sin C

■

You should be able to use the Law of Sines to solve an oblique triangle for the remaining three parts, given: (a) Two angles and any side (AAS or ASA) (b) Two sides and an angle opposite one of them (SSA) 1. If A is acute and h ⫽ b sin A: (a) a < h, no triangle is possible. (b) a ⫽ h or a > b, one triangle is possible. (c) h < a < b, two triangles are possible. 2. If A is obtuse and h ⫽ b sin A: (a) a ≤ b, no triangle is possible. (b) a > b, one triangle is possible.

■

The area of any triangle equals one-half the product of the lengths of two sides and the sine of their included angle. A ⫽ 12 ab sin C ⫽ 21ac sin B ⫽ 21bc sin A

Vocabulary Check 1. oblique

2.

1.

b sin B

C b

2. b

45° c

1 ac sin B 2

C

a = 20

30° A

3.

105°

a 40°

B A

c = 20

B

Given: A ⫽ 30⬚, B ⫽ 45⬚, a ⫽ 20 C ⫽ 180⬚ ⫺ A ⫺ B ⫽ 105⬚ a 20 sin 45⬚ 共sin B兲 ⫽ ⫽ 20冪2 ⬇ 28.28 b⫽ sin A sin 30⬚ c⫽

532

a 20 sin 105⬚ 共sin C兲 ⫽ ⬇ 38.64 sin A sin 30⬚

Given: B ⫽ 40⬚, C ⫽ 105⬚, c ⫽ 20 A ⫽ 180⬚ ⫺ B ⫺ C ⫽ 35⬚ a⫽

20 sin 35⬚ c ⬇ 11.88 共sin A兲 ⫽ sin 105⬚ sin C

b⫽

c 20 sin 40⬚ ⬇ 13.31 共sin B兲 ⫽ sin 105⬚ sin C

Section 6.1 3.

4.

C b

C b

a = 3.5

25°

a 135° 10°

35°

A

c

B

Law of Sines

A

c = 45

Given: A ⫽ 25⬚, B ⫽ 35⬚, a ⫽ 3.5

Given: B ⫽ 10⬚, C ⫽ 135⬚, c ⫽ 45

C ⫽ 180⬚ ⫺ A ⫺ B ⫽ 120⬚

A ⫽ 180⬚ ⫺ B ⫺ C ⫽ 35⬚

3.5 a 共sin B兲 ⫽ 共 sin 35⬚兲 ⬇ 4.75 b⫽ sin A sin 25⬚

a⫽

c 45 sin 35⬚ 共sin A兲 ⫽ ⬇ 36.50 sin 135⬚ sin C

b⫽

c 45 sin 10⬚ 共sin B兲 ⫽ ⬇ 11.05 sin C sin 135⬚

3.5 a 共sin C兲 ⫽ 共 sin 120⬚兲 ⬇ 7.17 sin A sin 25⬚

c⫽

5. Given: A ⫽ 36⬚, a ⫽ 8, b ⫽ 5 sin B ⫽

b sin A 5 sin 36⬚ ⫽ ⬇ 0.36737 ⇒ B ⬇ 21.55⬚ a 8

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 180⬚ ⫺ 36⬚ ⫺ 21.55 ⫽ 122.45⬚ c⫽

8 a 共sin C兲 ⫽ 共sin 122.45⬚兲 ⬇ 11.49 sin A sin 36⬚

6. Given: A ⫽ 60⬚, a ⫽ 9, c ⫽ 10 sin C ⫽

c sin A 10 sin 60⬚ ⫽ ⬇ 0.9623 ⇒ C ⬇ 74.21⬚ or C ⬇ 105.79⬚ a 9

Case 1

Case 2

C ⬇ 74.21⬚

C ⬇ 105.79⬚

B ⫽ 180⬚ ⫺ A ⫺ C ⬇ 45.79⬚

B ⫽ 180⬚ ⫺ A ⫺ C ⬇ 14.21⬚

b⫽

9 sin 45.79⬚ a 共 sin B兲 ⬇ ⬇ 7.45 sin 60⬚ sin A

7. Given: A ⫽ 102.4⬚, C ⫽ 16.7⬚, a ⫽ 21.6 B ⫽ 180⬚ ⫺ A ⫺ C ⫽ 60.9⬚

b⫽

9 sin 14.21⬚ a 共sin B兲 ⬇ ⬇ 2.55 sin 60⬚ sin A

8. Given: A ⫽ 24.3⬚, C ⫽ 54.6⬚, c ⫽ 2.68 B ⫽ 180⬚ ⫺ A ⫺ C ⫽ 101.1⬚

b⫽

21.6 a 共sin B兲 ⫽ 共sin 60.9⬚兲 ⬇ 19.32 sin 102.4⬚ sin A

a⫽

c 2.68 sin 24.3⬚ 共sin A兲 ⫽ ⬇ 1.35 sin C sin 54.6⬚

c⫽

21.6 a 共 sin C兲 ⫽ 共sin 16.7⬚兲 ⬇ 6.36 sin A sin 102.4⬚

b⫽

c 2.68 sin 101.1⬚ 共sin B兲 ⫽ ⬇ 3.23 sin 54.6⬚ sin C

9. Given: A ⫽ 83⬚ 20⬘, C ⫽ 54.6⬚, c ⫽ 18.1 B ⫽ 180⬚ ⫺ A ⫺ C ⫽ 180⬚ ⫺ 83⬚ 20⬘ ⫺ 54⬚ 36⬘ ⫽ 42⬚ 4⬘

10. Given: A ⫽ 5⬚ 40⬘, B ⫽ 8⬚ 15⬘, b ⫽ 4.8 C ⫽ 180⬚ ⫺ A ⫺ B ⫽ 166⬚ 5⬘

a⫽

18.1 c 共sin A兲 ⫽ 共sin 83⬚ 20⬘ 兲 ⬇ 22.05 sin C sin 54.6⬚

a⫽

b 4.8 sin 5⬚ 40⬘ 共sin A兲 ⫽ ⬇ 3.30 sin B sin 8⬚ 15⬘

b⫽

18.1 c 共sin B兲 ⫽ 共sin 42⬚ 4⬘兲 ⬇ 14.88 sin C sin 54.6⬚

c⫽

4.8 sin 166⬚ 5⬘ b ⬇ 8.05 共sin C兲 ⫽ sin B sin 8⬚ 15⬘

B

533

534

Chapter 6

Additional Topics in Trigonometry

11. Given: B ⫽ 15⬚ 30⬘ , a ⫽ 4.5, b ⫽ 6.8 sin A ⫽

a sin B 4.5 sin 15⬚ 30⬘ ⫽ ⬇ 0.17685 ⇒ A ⬇ 10⬚ 11⬘ b 6.8

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 180⬚ ⫺ 10⬚ 11⬘ ⫺ 15⬚ 30⬘ ⫽ 154⬚ 19⬘ c⫽

6.8 b 共sin C兲 ⫽ 共sin 154⬚ 19⬘ 兲 ⬇ 11.03 sin B sin 15⬚ 30⬘

12. Given: B ⫽ 2⬚ 45⬘, b ⫽ 6.2, c ⫽ 5.8 sin C ⫽

c sin B 5.8 sin 2⬚ 45⬘ ⫽ ⬇ 0.04488 ⇒ C ⬇ 2.57⬚ or 2⬚ 34⬘ b 6.2

A ⫽ 180 ⫺ B ⫺ C ⬇ 174.68⬚, or 174⬚ 41⬘ b 6.2 sin 174.68⬚ 共 sin A兲 ⬇ ⬇ 11.99 sin B sin 2⬚ 45⬘

a⫽

14. Given: A ⫽ 100⬚, a ⫽ 125, c ⫽ 10

13. Given: C ⫽ 145⬚, b ⫽ 4, c ⫽ 14 sin B ⫽

sin C ⫽

b sin C 4 sin 145⬚ ⫽ ⬇ 0.16388 ⇒ B ⬇ 9.43⬚ c 14

c sin A 10 sin 100⬚ ⫽ ⬇ 0.07878 ⇒ C ⬇ 4.52⬚ a 125

B ⫽ 180⬚ ⫺ A ⫺ C ⬇ 75.48⬚

A ⫽ 180⬚ ⫺ B ⫺ C ⬇ 180⬚ ⫺ 9.43⬚ ⫺ 145⬚ ⫽ 25.57⬚

b⫽

14 c 共sin A兲 ⬇ 共sin 25.57⬚兲 ⬇ 10.53 a⫽ sin 145⬚ sin C

a 125 sin 75.48⬚ 共sin B兲 ⬇ ⬇ 122.87 sin 100⬚ sin A

15. Given: A ⫽ 110⬚ 15⬘ , a ⫽ 48, b ⫽ 16 sin B ⫽

b sin A 16 sin 110⬚ 15⬘ ⫽ ⬇ 0.31273 ⇒ B ⬇ 18⬚ 13⬘ a 48

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 180⬚ ⫺ 110⬚ 15⬘ ⫺ 18⬚ 13 ⬘ ⫽ 51⬚ 32⬘ c⫽

48 a 共sin C兲 ⫽ 共sin 51⬚ 32⬘ 兲 ⬇ 40.06 sin A sin 110⬚ 15⬘

16. Given: C ⫽ 85⬚ 20 ⬘, a ⫽ 35, c ⫽ 50 sin A ⫽

a sin C 35 sin 85⬚ 20⬘ ⬇ 0.6977 ⇒ A ⬇ 44.24⬚, or 44⬚ 14⬘ ⫽ 50 c

B ⫽ 180⬚ ⫺ A ⫺ C ⬇ 50.43⬚, or 50⬚ 26⬘ b⫽

C sin B 50 sin 50.43⬚ ⬇ 38.67 ⬇ sin 85⬚ 20⬘ sin C

17. Given: A ⫽ 55⬚, B ⫽ 42⬚, c ⫽

3 4

C ⫽ 180⬚ ⫺ A ⫺ B ⫽ 83⬚

18. Given: B ⫽ 28⬚, C ⫽ 104⬚, a ⫽ 3 A ⫽ 180⬚ ⫺ B ⫺ C ⫽ 48⬚ 5

a⫽

0.75 c 共sin A兲 ⫽ 共sin 55⬚ 兲 ⬇ 0.62 sin C sin 83⬚

b⫽

a sin B 38 sin 28⬚ ⫽ ⬇ 2.29 sin A sin 48⬚

b⫽

0.75 c 共sin B兲 ⫽ 共sin 42⬚ 兲 ⬇ 0.51 sin C sin 83⬚

c⫽

a sin C 38 sin 104⬚ ⬇ 4.73 ⫽ sin 48⬚ sin A

5

5 8

Section 6.1 19. Given: A ⫽ 110⬚, a ⫽ 125, b ⫽ 100 sin B ⫽

Law of Sines

535

20. Given: a ⫽ 125, b ⫽ 200, A ⫽ 110⬚

b sin A 100 sin 110⬚ ⬇ 0.75175 ⇒ B ⬇ 48.74⬚ ⫽ 125 a

No triangle is formed because A is obtuse and a < b.

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 21.26⬚ c⫽

a sin C 125 sin 21.26⬚ ⬇ 48.23 ⬇ sin 110⬚ sin A 22. Given: A ⫽ 76⬚, a ⫽ 34, b ⫽ 21

21. Given: a ⫽ 18, b ⫽ 20, A ⫽ 76⬚ h ⫽ 20 sin 76⬚ ⬇ 19.41

sin B ⫽

Since a< h, no triangle is formed.

b sin A 21 sin 76⬚ ⬇ 0.5993 ⇒ B ⬇ 36.82⬚ ⫽ 34 a

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 67.18⬚ c⫽

a sin C 34 sin 67.18⬚ ⬇ ⬇ 32.30 sin A sin 76⬚

23. Given: A ⫽ 58⬚, a ⫽ 11.4, c ⫽ 12.8 sin B ⫽

b sin A 12.8 sin 58⬚ ⬇ 0.9522 ⇒ B ⬇ 72.21⬚ or B ⬇ 107.79⬚ ⫽ 11.4 a

Case 1

Case 2

B ⬇ 72.21⬚

B ⬇ 107.79⬚

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 49.79⬚

C ⫽ 180⬚ ⫺ A ⫺ B ⬇ 14.21⬚

11.4 sin 49.79⬚ a 共sin C兲 ⬇ ⬇ 10.27 c⫽ sin 58⬚ sin A

c⫽

11.4 sin 14.21⬚ a 共sin C兲 ⬇ ⬇ 3.30 sin 58⬚ sin A

25. Given: A ⫽ 36⬚, a ⫽ 5

24. Given: a ⫽ 4.5, b ⫽ 12.8, A ⫽ 58⬚ h ⫽ 12.8 sin 58⬚ ⬇ 10.86

(a) One solution ifb ≤ 5 or b ⫽

Since a < h, no triangle is formed.

5 sin 36⬚

5 (b) Two solutions if 5 < b < sin 36⬚ (c) No solution if b >

27. Given: A ⫽ 10⬚, a ⫽ 10.8

26. Given: A ⫽ 60⬚, a ⫽ 10 (a) One solution if b ≤ 10 orb ⫽ (b) Two solutions if 10 < b < (c) No solutions if b >

10 . sin 60⬚

10 . sin 60⬚

10 . sin 60⬚

315.6 sin 88⬚

(b) Two solutions if 10.8 < b <

10.8 sin 10⬚

10.8 sin 10⬚

10.8 sin 10⬚

1 1 29. Area ⫽ ab sin C ⫽ 共4兲共6 兲 sin 120⬚ ⬇ 10.4 2 2

(a) One solution if b ≤ 315.6 or b ⫽ (b) Two solutions if 315.6 < b <

(a) One solution ifb ≤ 10.8 or b ⫽

(c) No solution if b >

28. Given: A ⫽ 88⬚, a ⫽ 315.6

(c) No solutions if b >

5 sin 36⬚

315.6 sin 88⬚

315.6 sin 88⬚

536

Chapter 6

Additional Topics in Trigonometry

30. Area ⫽ 21ac sin B ⫽ 21共 62 兲共20兲 sin 130⬚ ⬇ 474.9

31. Area ⫽ 21bc sin A ⫽ 2 共57兲共85 兲 sin 43⬚ 45⬘ ⬇ 1675.2

32. A ⫽ 5⬚ 15⬘, b ⫽ 4.5, c ⫽ 22

1 33. Area ⫽ 2 ac sin B ⫽ 2共 105兲共 64兲sin共72⬚30⬘兲 ⬇ 3204.5

1

1

1

Area ⫽ 2 bc sin A ⫽

共12兲 共4.5 兲共22兲 sin 5.25⬚ ⬇ 4.5

34. C ⫽ 84⬚ 30⬘, a ⫽ 16, b ⫽ 20

35. C ⫽ 180⬚ ⫺ 23⬚ ⫺ 94⬚ ⫽ 63⬚

1 Area ⫽ ab sin C 2 ⫽

h⫽

冢2冣 共16 兲共 20兲 sin 84.5⬚ ⬇ 159.3 1

36. (a)

37. 20° h

32° 16

42⬚ ⫺ ␪ ⬇ 25.9⬚

␪ ⬇ 16.1⬚

16 h ⫽ sin 32⬚ sin 70⬚

(c) h ⫽

38.

16 sin 32⬚ ⬇ 9.0 meters sin 70⬚ 39. Given: c ⫽ 100

N W

E

A ⫽ 74⬚ ⫺ 28⬚ ⫽ 46⬚,

S

B ⫽ 180⬚ ⫺ 41⬚ ⫺ 74⬚ ⫽ 65⬚,

Elgin C a = 720 km

C ⫽ 180⬚ ⫺ 46⬚ ⫺ 65⬚ ⫽ 69⬚ b = 500 km 44°

46° B Canton

A Naples

Given: A ⫽ 46⬚, a ⫽ 720, b ⫽ 500 sin B ⫽

b sin A 500 sin 46⬚ ⬇ 0.50 ⇒ B ⬇ 30⬚ ⫽ 720 a

The bearing from C to B is 240⬚. 40.

3000 ft

r 40°

s

r

(b) r ⫽

sin共42⬚ ⫺ ␪兲 sin 48⬚ ⫽ 10 17 sin 共42⬚ ⫺ ␪兲 ⬇ 0.43714

70°

12°

(b)

35 共sin 23⬚兲 ⬇ 15.3 meters sin 63⬚

3000 sin 关12共 180⬚ ⫺ 40⬚兲兴 ⬇ 4385.71 feet sin 40⬚

冢180 冣4385.71 ⬇ 3061.80 feet

(c) s ⬇ 40

␲

A

100 46° 65° 69° C

100 c 共sin A兲 ⫽ 共 sin 46⬚兲 ⬇ 77 meters a⫽ sin C sin 69⬚

B

Section 6.1 41. (a)

42. Given: A ⫽ 15⬚, B ⫽ 135⬚, c ⫽ 30

18.8° 17.5°

C ⫽ 180⬚ ⫺ A ⫺ B ⫽ 30⬚

z

x

537

Law of Sines

From Pine Knob: y

9000 ft

Not drawn to scale

(b)

b⫽

9000 x ⫽ sin 17.5⬚ sin 1.3⬚

c sin B 30 sin 135⬚ ⬇ 42.4 kilometers ⫽ sin 30⬚ sin C

From Colt Station:

x ⬇ 119,289.1261 feet ⬇ 22.6 miles

a⫽

c sin A 30 sin 15⬚ ⬇ 15.5 kilometers ⫽ sin 30⬚ sin C

x y ⫽ sin 71.2⬚ sin 90⬚

(c)

B

y ⫽ x sin 71.2⬚ ⬇ 119,289.1261 sin 71.2⬚

a

30 65° 70°

80° c = 15° 65°

⬇ 112,924.963 feet ⬇ 21.4 miles (d) z ⫽ x sin 18.8⬚ ⬇ 119,289.1261 sin 18.8⬚

C

b

A

⬇ 38,443 feet ⬇ 7.3 miles

43.

In 15 minutes the boat has traveled

10 mi 4

70°

63°

20°

共10 mph 兲共 14 hr兲 ⫽ 10 4 miles.

27°

y

d

θ

␪ ⫽ 180⬚ ⫺ 20⬚ ⫺ 共90⬚ ⫹ 63⬚兲 ␪ ⫽ 7⬚ 10兾4 y ⫽ sin 7⬚ sin 20⬚ y ⬇ 7.0161 sin 27⬚ ⫽

d 7.0161

d ⬇ 3.2 miles

44. (a) sin ␣ ⫽ (c)

5.45 ⬇ 0.0934 ⇒ ␣ ⬇ 5.36⬚ 58.36

(b)

d 58.36 or ⫽ sin ␪ sin共84.64⬚ ⫺ ␪兲 d⫽

sin ␤ sin ␪ ⫽ 58.36 d sin ␤ ⫽

58.36 sin共 84.64⬚ ⫺ ␪ 兲 sin ␪

␤ ⫽ sin⫺1 (d)

45. True. If one angle of a triangle is obtuse, then there is less than 90⬚ left for the other two angles, so it cannot contain a right angle. It must be oblique.

d sin ␪ 58.36

冢 58.36 冣 d sin ␪

␪

10 ⬚

20 ⬚

30 ⬚

40 ⬚

50 ⬚

60 ⬚

d

324.1

154.2

95.19

63.80

43.30

28.10

46. False. Two sides and one opposite angle do not necessarily determine a unique triangle.

538

Chapter 6

47. (a)

Additional Topics in Trigonometry

sin ␣ sin ␤ ⫽ 18 9 sin ␣ ⫽ 0.5 sin ␤

␣ ⫽ arcsin共0.5 sin ␤ 兲 (b)

Domain: 0 < ␤ < ␲

1

Range: 0 < ␣ ≤ 0

␲ 6

␲ 0

␥ ⫽ ␲ ⫺ ␣ ⫺ ␤ ⫽ ␲ ⫺ ␤ ⫺ arcsin共0.5 sin ␤兲

(c)

18 c ⫽ sin ␤ sin ␥ c⫽ (d)

18 sin ␥ 18 sin 关␲ ⫺ ␤ ⫺ arcsin共 0.5 sin ␤ 兲兴 ⫽ sin ␤ sin ␤

27

Domain: 0 < ␤ < ␲ Range: 9 < c < 27 0

␲ 0

(e)

␤

0.4

0.8

1.2

1.6

2.0

2.4

2.8

As ␤ → 0, c → 27

␣

0.1960

0.3669

0.4848

0.5234

0.4720

0.3445

0.1683

As ␤ → ␲, c → 9

c

25.95

23.07

19.19

15.33

12.29

10.31

9.27

48. (a) A ⫽

冢

␪ 1 共30 兲共20兲 sin ␪ ⫹ 2 2

␪ 3␪ ⫺ 80 sin ⫺ 120 sin ␪ 2 2

(a)

⫽ 300 sin

(a)

⫽ 20 15 sin

冤

(b)

冣 ⫺ 21 共8兲共20兲 sin 2␪ ⫺ 21 共 8兲共30兲 sin ␪ 20 cm θ 2

冥

␪ 3␪ ⫺ 4 sin ⫺ 6 sin ␪ 2 2

8 cm θ 30 cm

(c) Domain: 0 ≤ ␪ ≤ 1.6690

170

The domain would increase in length and the area would have a greater maximum value if the 8-centimeter line segment were decreased. 0

1.7 0

49. sin x cot x ⫽ sin x

51. 1 ⫺ sin2

cos x ⫽ cos x sin x

冢 2␲ ⫺ x 冣 ⫽ 1 ⫺ cos

2

x ⫽ sin2 x

53. 6 sin 8 ␪ cos 3 ␪ ⫽ 共6 兲共12 兲关sin 共8 ␪ ⫹ 3 ␪兲 ⫹ sin 共8 ␪ ⫺ 3␪ 兲兴 ⫽ 3 共sin 11␪ ⫹ sin 5␪兲

50. tan x cos x sec x ⫽ tan x cos x

52. 1 ⫹ cot 2

1 ⫽ tan x cos x

冢␲2 ⫺ x冣 ⫽ 1 ⫹ tan x ⫽ sec 2

2

x

54. 2 cos 5␪ sin 2␪ ⫽ 2 ⭈ 12关sin共 5␪ ⫹ 2␪ 兲 ⫺ sin共5 ␪ ⫺ 2 ␪ 兲兴 ⫽ sin 7␪ ⫺ sin 3␪

Section 6.2

Section 6.2

■

■

Law of Cosines

If ABC is any oblique triangle with sides a, b, and c, the following equations are valid. (a) a2 ⫽ b 2 ⫹ c2 ⫺ 2bc cos A

or

cos A ⫽

b 2 ⫹ c 2 ⫺ a2 2bc

(b) b 2 ⫽ a 2 ⫹ c2 ⫺ 2ac cos B

or

cos B ⫽

a2 ⫹ c2 ⫺ b 2 2ac

(c) c2 ⫽ a 2 ⫹ b2 ⫺ 2ab cos C or

cos C ⫽

a2 ⫹ b2 ⫺ c2 2ab

You should be able to use the Law of Cosines to solve an oblique triangle for the remaining three parts, given: (a) Three sides (SSS) (b) Two sides and their included angle (SAS)

■

Law of Cosines

Given any triangle with sides of length a, b, and c, the area of the triangle is Area ⫽ 冪s 共s ⫺ a 兲共s ⫺ b兲共 s ⫺ c兲, where s ⫽

a⫹b⫹c . 2

(Heron’s Formula)

Vocabulary Check 1. Cosines

2. b 2 ⫽ a2 ⫹ c2 ⫺ 2ac cos B

3. Heron’s Area

1. Given: a ⫽ 7, b ⫽ 10, c ⫽ 15 cos C ⫽

a2 ⫹ b2 ⫺ c2 49 ⫹ 100 ⫺ 225 ⫽ ⬇ ⫺0.5429 ⇒ C ⬇ 122.88⬚ 2共7兲共 10兲 2ab

sin B ⫽

b sin C 10 sin 122.88⬚ ⫽ ⬇ 0.5599 ⇒ B ⬇ 34.05⬚ c 15

A ⬇ 180⬚ ⫺ 34.05⬚ ⫺ 122.88⬚ ⬇ 23.07⬚ 2. Given: a ⫽ 8, b ⫽ 3, c ⫽ 9 cos C ⫽

82 ⫹ 32 ⫺ 92 a2 ⫹ b 2 ⫺ c2 ⫽ ⬇ ⫺0.16667 ⇒ C ⫽ 99.59⬚ 2ab 2 共8 兲共3 兲

sin A ⫽

a sin C 8 sin 99.59⬚ ⫽ ⫽ 0.8765 ⇒ A ⫽ 61.22⬚ 9 c

B ⬇ 180⬚ ⫺ 61.22⬚ ⫺ 99.59⬚ ⬇ 19.19⬚ 3. Given: A ⫽ 30⬚, b ⫽ 15, c ⫽ 30 a2 ⫽ b 2 ⫹ c2 ⫺ 2bc cos A ⫽ 225 ⫹ 900 ⫺ 2共15兲共 30兲 cos 30⬚ ⬇ 345.5771 a ⬇ 18.59 cos C ⫽

a2 ⫹ b 2 ⫺ c2 共18.59 兲2 ⫹ 152 ⫺ 30 2 ⬇ ⬇ ⫺0.5907 ⇒ C ⬇ 126.21⬚ 2共 18.59兲共15 兲 2ab

B ⬇ 180⬚ ⫺ 30⬚ ⫺ 126.21⬚ ⫽ 13.79⬚

539

540

Chapter 6

Additional Topics in Trigonometry

4. Given: C ⫽ 105⬚, a ⫽ 10, b ⫽ 4.5 c2 ⫽ a 2 ⫹ b 2 ⫺ 2ab cos C ⫽ 10 2 ⫹ 4.5 2 ⫺ 2 共 10兲共4.5 兲 cos 105⬚ ⬇ 143.5437 ⇒ c ⬇ 11.98 a2 ⫹ c2 ⫺ b2 102 ⫹ 共12.0兲 2 ⫺ 共 4.5兲2 ⬇ ⬇ 0.93187 ⇒ B ⬇ 21.27⬚ 2 共10 兲共12.0兲 2ac

cos B ⫽

A ⫽ 180⬚ ⫺ 105⬚ ⫺ 21.27⬚ ⬇ 53.73⬚ 5. a ⫽ 11, b ⫽ 14, c ⫽ 20 cos C ⫽

a2 ⫹ b2 ⫺ c2 121 ⫹ 196 ⫺ 400 ⫽ ⬇ ⫺0.2695 ⇒ C ⬇ 105.63⬚ 2ab 2共11兲共14 兲

sin B ⫽

b sin C 14 sin 105.63⬚ ⫽ ⬇ 0.6741 ⇒ B ⬇ 42.38⬚ c 20

A ⬇ 180⬚ ⫺ 42.38⬚ ⫺ 105.63⬚ ⬇ 31.99⬚ 6. Given: a ⫽ 55, b ⫽ 25, c ⫽ 72 cos C ⫽

552 ⫹ 252 ⫺ 722 a 2 ⫹ b 2 ⫺ c2 ⫽ ⬇ ⫺0.5578 ⇒ C ⬇ 123.91⬚ 2ab 2 共55 兲共25 兲

cos A ⫽

b2 ⫹ c2 ⫺ a2 252 ⫹ 72 2 ⫺ 55 2 ⫽ ⬇ 0.7733 ⇒ A ⬇ 39.35⬚ 2 共25兲共 72兲 2bc

B ⫽ 180⬚ ⫺ 123.91⬚ ⫺ 39.35⬚ ⬇ 16.74⬚ 7. Given: a ⫽ 75.4, b ⫽ 52, c ⫽ 52 cos A ⫽

b2 ⫹ c 2 ⫺ a2 52 2 ⫹ 522 ⫺ 75.42 ⫽ ⫺0.05125 ⇒ A ⬇ 92.94⬚ ⫽ 2共52兲共52兲 2bc

sin B ⫽

b sin A 52共 0.9987兲 ⬇ ⬇ 0.68876 ⇒ B ⬇ 43.53⬚ a 75.4

C ⫽ B ⬇ 43.53⬚ 8. Given: a ⫽ 1.42, b ⫽ 0.75, c ⫽ 1.25 cos A ⫽

b2 ⫹ c ⫺ a2 共0.75兲2 ⫹ 共1.25 兲2 ⫺ 共1.42 兲2 ⫽ ⫽ 0.05792 ⇒ A ⬇ 86.68⬚ 2bc 2共0.75兲共1.25兲

cos B ⫽

共1.42 兲2 ⫹ 共1.25兲2 ⫺ 共0.75兲2 a2 ⫹ c 2 ⫺ b2 ⬇ 0.84969 ⇒ B ⬇ 31.82⬚ ⫽ 2ac 2 共1.42兲共 1.25兲

2

C ⫽ 180⬚ ⫺ 86.68⬚ ⫺ 31.82⬚ ⬇ 61.50⬚ 9. Given: A ⫽ 135⬚, b ⫽ 4, c ⫽ 9 a2 ⫽ b 2 ⫹ c2 ⫺ 2bc cos A ⫽ 16 ⫹ 81 ⫺ 2共4 兲共9兲cos 135⬚ ⬇ 147.9117 ⇒ a ⬇ 12.16 sin B ⫽

b sin A 4 sin 135⬚ ⫽ ⬇ 0.2326 ⇒ B ⬇ 13.45⬚ a 12.16

C ⬇ 180⬚ ⫺ 135⬚ ⫺ 13.45⬚ ⬇ 31.55⬚ 10. Given: A ⫽ 55⬚, b ⫽ 3, c ⫽ 10 a2 ⫽ b 2 ⫹ c 2 ⫺ 2bc cos A ⫽ 3 2 ⫹ 102 ⫺ 2 共 3兲共10 兲 cos 55⬚ ⬇ 74.585 ⇒ a ⬇ 8.64 sin B ⫽

b sin A 3 sin 55⬚ ⬇ 0.2846 ⇒ A ⬇ 16.53⬚ ⬇ 8.636 a

C ⬇ 180⬚ ⫺ 16.53⬚ ⫺ 55⬚ ⬇ 108.47⬚

Section 6.2 11. Given: B ⫽ 10⬚ 35 ⬘, a ⫽ 40, c ⫽ 30 b2 ⫽ a 2 ⫹ c2 ⫺ 2ac cos B ⫽ 1600 ⫹ 900 ⫺ 2共40兲共 30兲cos 10⬚ 35 ⬘ ⬇ 140.8268 ⇒ b ⬇ 11.87 sin C ⫽

c sin B 30 sin 10⬚ 35⬘ ⬇ 0.4642 ⇒ C ⬇ 27.66⬚ ⬇ 27⬚ 40⬘ ⫽ 11.87 b

A ⬇ 180⬚ ⫺ 10⬚ 35⬘ ⫺ 27⬚ 40⬘ ⫽ 141⬚ 45 ⬘ 12. Given: B ⫽ 75⬚ 20⬘, a ⫽ 6.2, c ⫽ 9.5 b2 ⫽ a 2 ⫹ c 2 ⫺ 2ac cos B ⫽ 共6.2 兲2 ⫹ 共9.5 兲2 ⫺ 2 共6.2兲共9.5 兲 cos 75⬚ 20⬘ ⬇ 98.8636 ⇒ b ⬇ 9.94 sin A ⫽

a sin B 6.2 sin 75⬚ 20⬘ ⬇ 0.6034 ⇒ A ⬇ 37.1⬚, or 37⬚ 6 ⬘ ⬇ 9.94 b

C ⬇ 180⬚ ⫺ 75⬚ 20⬘ ⫺ 37⬚ 6⬘ ⬇ 67⬚ 34⬘ 13. Given: B ⫽ 125⬚ 40⬘, a ⫽ 32, c ⫽ 32 b 2 ⫽ a 2 ⫹ c 2 ⫺ 2ac cos B ⬇ 322 ⫹ 322 ⫺ 2共 32兲共32兲 cos 125⬚ 40 ⬘ ⬇ 3242.1888 ⇒ b ⬇ 56.94 A ⫽ C ⇒ 2A ⫽ 180⬚ ⫺ 125⬚ 40⬘ ⫽ 54⬚ 20⬘ ⇒ A ⫽ C ⫽ 27⬚ 10⬘ 14. Given: C ⫽ 15⬚ 15⬘, a ⫽ 6.25, b ⫽ 2.15 c2 ⫽ a 2 ⫹ b 2 ⫺ 2ab cos C ⫽ 共6.25兲2 ⫹ 共2.15兲 2 ⫺ 2共 6.25兲共2.15兲 cos 15⬚ 15⬘ ⬇ 17.7563 ⇒ c ⬇ 4.21 cos A ⫽

b2 ⫹ c 2 ⫺ a2 共2.15 兲2 ⫹ 共4.2138 兲2 ⫺ 共6.25兲2 ⬇ ⬇ ⫺0.9208 ⇒ A ⬇ 157.04⬚ or 157⬚ 2⬘ 2 共2.15兲共 ...