chapter 6 analytic trigonometry...2013/07/31 · chapter 6 analytic trigonometry ... a

Chapter 6 Analytic Trigonometry

Copyright © 2014 Pearson Education, Inc. 705

Section 6.1

Check Point Exercises

1. 1 sin

csc tansin cos

xx x

x x

1

cossec

xx

We worked with the left side and arrived at the right side. Thus, the identity is verified.

2. cos

cos cot sin cos sinsin

xx x x x x

x

2

2

2 2

2 2

cossin

sin

cos sinsin

sin sin

cos sin

sin sin

cos sin

sin1

sincsc

xx

x

x xx

x x

x x

x x

x x

x

xx


3. 2 22

3

sin sin cos sin 1 cos

sin sin

sin

x x x x x

x x

x


4. 1 cos 1 cos

sin sin sincsc cot

θ θθ θ θ

θ θ


5. sin 1 cos

1 cos sin

x x

x x

2 2

2 2

sin (sin ) (1 cos )(1 cos )

(1 cos )sin sin (1 cos )

sin 1 2cos cos

(1 cos )sin (1 cos )sin

sin cos 2cos 1

(1 cos )sin

1 1 2cos

(1 cos )sin

2 2cos

(1 cos )sin

2 1 cos

(1 cos )sin

2

sin2cs

x x x x

x x x x

x x x

x x x x

x x x

x x

x

x x

x

x x

x

x x

x

c x


6.

2

2

cos cos 1 sin

1 sin (1 sin ) 1 sin

cos (1 sin )

1 sincos (1 sin )

cos1 sin

cos

x x x

x x x

x x

xx x

xx

x



706 Copyright © 2014 Pearson Education, Inc.

7. sec csc( )

sec csc

x x

x x

sec csc

sec csc1 1

cos sin1 1

cos sinsin cos

cos sin cos sin1

cos sinsin cos

cos sin1

cos sinsin cos cos sin

cos sin 1sin cos

x x

x x

x x

x xx x

x x x x

x xx x

x x

x xx x x x

x xx x


8. Left side:

2

1 1

1 sin 1 sin1(1 sin ) 1(1 sin )

(1 sin )(1 sin ) (1 sin )(1 sin )

1 sin 1 sin

(1 sin )(1 sin )

2

(1 sin )(1 sin )

2

1 sin

θ θθ θ

θ θ θ θθ θθ θ

θ θ

θ

Right side: 2

22

2 2

2 2

2 2

2

2 2

sin2 2 tan 2 2

cos

2cos 2sin

cos cos

2cos 2sin

cos2 2

cos 1 sin

θθθ

θ θθ θθ θ

θ

θ θ

The identity is verified because both sides are equal

to 2

2.

1 sin θ

Concept and Vocabulary Check 6.1

1. complicated; other

2. sines; cosines

3. false

4. (csc 1)(csc 1)x x

5. identical/the same

Exercise Set 6.1

1. 1

sin sec sincos

sin

costan

x x xx

x

xx

2. 1

2cos csc cossin

x x xx

cos

sincot

x

xx

3. tan( ) cos tan cos

sincos

cossin

x x x x

xx

xx

4. cot( ) sin cot sin

cossin

sincos

x x x x

xx

xx

5. sin 1

tan csc cos coscos sin1

xx x x x

x x

6. cos 1

cot sec sin sinsin cos1

xx x x x

x x

7. 2 2sec sec sin sec (1 sin )

1 2coscoscos

x x x x x

xxx

Section 6.1 Verifying Trigonometric Identities


8. 2 2csc csc cos csc (1 cos )

1 2sinsinsin

x x x x x

xxx

9. 2 2 2 22 2

2

cos sin 1 sin sin

1 sin sin

1 2sin

x x x x

x x

x

10. 2 2 2 2cos sin cos (1 cos )2 2cos 1 cos

22cos 1

x x x x

x x

x

11. 1

csc sin sinsin

θ θ θθ

2

2

2

1 sin

sin sin

1 sin

sin

cos

sincos

cossincot cos

θθ θ

θθθ

θθ θθθ θ

12. sin cos

tan cotcos sin

θ θθ θθ θ

sin sin cos cos

cos sin sin cos2 2sin cos

cos sin cos sin2 2sin cos

cos sin1

cos sin1 1

cos sinsec csc

θ θ θ θθ θ θ θ

θ θθ θ θ θθ θ

θ θ

θ θ

θ θθ θ

13.

sin costan cot cos sin

1cscsin

θ θθ θ θ θ

θθ

11

sin1

1sin

sin1

1sin

θ

θθ

θ

14.

cos 1cos sec 1 cos

coscotsin

θθ θ θ

θθθ

1cos

sincos

1sin

sin1

costan

θθ

θθ

θθ

θ

15. 2 2 2 2sin (1 cot ) sin (csc )θ θ θ θ

2

2

1sin

sin1

θθ

16. 2 2 2 2cos (1 tan ) cos (sec )θ θ θ θ

2

2

1cos

cos1

θθ

17. 2 21 cos sin

cos cossin

sincos

sin tan

t t

t tt

tt

t t

18.

2 21 sin cos

sin sincos

cossin

cos cot

t t

t tt

tt

t t



19. 2 2

2

2

1csc sin

coscotsin1 cos

sinsin1 sin

cossin1 1

sin coscsc sec

t tttt

t

ttt

tt

t tt t

20. 2 2

2

2

1sec cos

sintancos

1 sin

coscos1 cos

sincos1 1

cos sinsec csc

t tttt

t

ttt

tt

t tt t

21. 2 2

2

tan sec 1

sec sec

sec 1

sec secsec cos

t t

t t

t

t tt t

22. 2 2

2

cot csc 1

csc csc

csc 1

csc csccsc sin

t t

t t

t

t tt t

23. 1 cos 1 cos

sin sin sincsc cot

θ θθ θ θ

θ θ

24. 1 sin 1 sin

cos cos cossec tan

θ θθ θ θ

θ θ

25.

2 2

sin cos sin cos1 1csc sec

sin cos1 1

sin cossin cos

sin cossin cos

1 1

sin cos

1

t t t t

t tt t

t tt t

t tt t

t t

26. sin cos sin cossin costan cotcos sin

sin cossin cos

cos sincos sin

sin cossin cos

cos sin

sin cos

t t t tt tt tt t

t tt t

t tt t

t tt t

t t

t t

27. costan1 sin

sin cos

cos 1 sinsin 1 sin cos cos

cos 1 sin 1 sin cos2 2sin sin cos

cos (1 sin ) cos (1 sin )2 2sin sin cos

cos (1 sin )

1 sin

cos (1 sin )

1

cossec

tt

tt t

t tt t t t

t t t t

t t t

t t t t

t t t

t t

t

t t

tt



28. sin cos sincot1 cos sin 1 cos

cos 1 cos sin sin

sin 1 cos 1 cos sin2 2cos cos sin

sin (1 cos ) sin (1 cos )2 2cos cos sin

sin (1 cos )

cos 1

sin (1 cos )

1

sincsc

t t tt

t t tt t t t

t t t t

t t t

t t t t

t t t

t t

t

t t

tt

29. 2 2

2

2

2

2

sin sin 1 cos1 1

1 cos 1 cos 1 cos

sin (1 cos )1

1 cos

sin (1 cos )1

sin1 1 cos

cos

x x x

x x x

x x

x

x x

xx

x

30. 2 2

2

2

2

2

cos cos 1 sin1 1

1 sin 1 sin 1 sin

cos (1 sin )1

1 sin

cos (1 sin )1

cos1 1 sin

sin

x x

x x x

x x

x

x x

xx

x

31.

2

2

cos 1 sin

1 sin coscos 1 sin 1 sin

1 sin 1 sin coscos (1 sin ) 1 sin

cos1 sincos (1 sin ) 1 sin

coscos1 sin 1 sin

cos cos2

cos1

2cos

2sec

x x

x xx x x

x x xx x x

xxx x x

xxx x

x x

x

xx

32.

2

2

2

sin cos 1

cos 1 sinsin cos 1 cos 1

cos 1 cos 1 sinsin (cos 1) cos 1

sincos 1sin (cos 1) cos 1

sinsinsin (1 cos ) cos 1

sinsin1 cos cos 1

sin sin0

sin0

x x

x xx x x

x x xx x x

xxx x x

xxx x x

xxx x

x x

x

33. 2 2 2 2

2 2 2

22

2 2

22

2 2

2 2

sec csc (1 tan )csc

csc tan csc

sin 1csc

cos sin1

csccos

csc sec

sec csc

x x x x

x x x

xx

x x

xx

x x

x x



34. 2 2

2

2

2

2

csc sec (1 cot )sec

sec cot sec

cos 1sec

cossincos

secsin

1 cossec

sin sinsec csc cot

x x x x

x x x

xx

xxx

xx

xx

x xx x x

35.

1 1sec csc cos sin

1 1sec csccos sin

x x x xx x

x x

1 1sincos sin

1 1 sincos sinsin

1cossin

1costan 1

tan 1

xx xx

x xx

xx

xx

x

36.

1 1csc sec sin cos

1 1csc secsin cos

x x x xx x

x x

1 1cossin cos

1 1 cossin coscos

1sincos

1sincot 1

cot 1

xx xx

x xx

xx

xx

x

37. 2 2sin cos (sin cos )(sin cos )

sin cos sin cossin cos

x x x x x x

x x x xx x

38. 2 2tan cot (tan cot )(tan cot )

tan cot tan cottan cot

x x x x x x

x x x xx x

39. 2 2 2 2

2

tan 2 sin 2 cos 2 tan 2 1

sec 2

x x x x

x

40. 2 2 2 2

2

cot 2 cos 2 sin 2 cot 2 1

csc 2

x x x x

x

41.

2

sin 2 cos 2tan 2 cot 2 cos 2 sin 2

1csc 2sin

sin 2 sin 2 cos 2 cos 2

cos 2 sin 2 sin 2 cos 21

sin

sin 2 cos2

cos 2 sin 21

sin 21 1

cos 2 sin 2 sin 21 sin 2

cos 2 sin 2 11

sec 2cos 2

θ θθ θ θ θ

θθ


θθ θθ θ

θ

θ θ θθ

θ θ

θθ



42.

2 2

2 2

sin 2 cos 2tan 2 cot 2 cos 2 sin 2

1sec 2cos 2

sin 2 sin 2 cos 2 cos 2

cos 2 sin 2 sin 2 cos 21

cos2

sin 2 cos 2

cos 2 sin 2 cos 2 sin 21

cos 2

sin 2 cos 2

cos2 sin 21

cos 21 1

cos 2 sin 2 cos21 cos 2

cos 2 sin 2

θ θθ θ θ θ

θθ


θθ θ

θ θ θ θ

θθ θ

θ θ

θ

θ θ θ

θ θ

1

1

sin 2csc 2

θ

θθ

43. sin sin

tan tan cos coscos cossin sin1 tan tan cos cos1cos cos

sin cos cos sin

cos cos sin sin

x y

x y x yx yx yx y x yy y

x y x y

x y x y

44. cos cos

cot cot sin sincos cos1 cot cot 1sin sin

cos cos

sin sinsin sincos cos sin sin1sin sin

cos sin sin cos sin sin

sin 1 sin 1cos cos sin sin

sin sinsin sin 1

cos sin sin co

x y

x y x yx yx yx y

x y

x yx yx y x yx y

x x y y x y

x yx y x y

x yx y

x y x

ssin sin cos cos

y

x y x y



45. Left side: 2

2

2

2

2

1 sin(sec tan )

cos cos

1 sin

cos

(1 sin )

cos

xx x

x x

x

x

x

x

Right side:

2

2

2

2

1 sin 1 sin 1 sin

1 sin 1 sin 1 sin

(1 sin )

1 sin

(1 sin )

cos

x x x

x x x

x

x

x

x

The identity is verified because both sides are equal to 2

2

(1 sin )

cos

x

x

.

46. Left side: 2 2 2

22

1 cos 1 cos (1 cos )(csc cot )

sin sin sin sin

x x xx x

x x x x

Right side: 2 2

2 2

1 cos 1 cos 1 cos (1 cos ) (1 cos )

1 cos 1 cos 1 cos 1 cos sin

x x x x x

x x x x x

The identity is verified because both sides are equal to2

2

(1 cos )

sin

x

x

.

47.

2

2

tan tan sec 1

sec 1 sec 1 sec 1tan (sec 1)

sec 1tan (sec 1)

tansec 1

tan

t t t

t t tt t

tt t

tt

t

48.

2

2

cot cot csc 1

csc 1 csc 1 csc 1cot (csc 1)

csc 1cot (csc 1)

cotcsc 1

cot

t t t

t t tt t

tt t

tt

t



49. Left side:

2

2

2

2

1 cos 1 cos 1 cos

1 cos 1 cos 1 cos

(1 cos )

1 cos

(1 cos )

sin

t t t

t t t

t

t

t

t

Right side: 2

2

2

2

2

1 cos(csc cot )

sin sin

1 cos

sin

(1 cos )

sin

tt t

t t

t

t

t

t

The identity is verified because both sides are equal to 2

2

(1 cos )

sin

t

t

.

50. Left side: 2cos 4cos 4 (cos 2)(cos 2)

cos 2 cos 2cos 2

t t t t

t tt

Right side: 2sec 1 2sec 1

sec sec sec2 cos

cos 2

t t

t t tt

t

The identity is verified because both sides are equal to cos 2t .

51.

4 4 2 2 2 2

2 2

2 2

2

cos sin cos sin cos sin

cos sin 1

1 sin sin

1 2sin

t t t t t t

t t

t t

t

52.

4 4 2 2 2 2

2 2

2 2

2

sin cos sin cos sin cos

sin cos 1

1 cos cos

1 2cos

t t t t t t

t t

t t

t



53. sin cos cos sin

sin cos

θ θ θ θθ θ

2 2

2 2

(sin cos )cos (cos sin )sin

cos sin cos sin

sin cos cos sin cos sin

sin cos

2sin cos cos sin

sin cos2sin cos 1

sin cos2sin cos 1

sin cos sin cos1 1

2sin cos

2 csc sec

2 sec csc

θ θ θ θ θ θθ θ θ θ

θ θ θ θ θ θθ θ

θ θ θ θθ θ

θ θθ θ

θ θθ θ θ θ

θ θθ θθ θ

54. sin cos

1 cot tan 1

θ θθ θ

2

sin coscos sin

1 1sin cossin cos

sin cos sin cos

sin sin cos cossin cos

sin cos sin cos

sin cossin cos sin cos

sin cossin cos

sin cossin cos

sin cos sin cos

sin

sin cos

θ θθ θθ θθ θ


θ θθ θ θ θ

θ θθ θ θ θθ θ

θ θθ θθ θ

θ θ θ θθ

θ

2

2 2

cos

sin cos

sin cos

sin cos(sin cos )(sin cos )

sin cossin cos

θθ θ θ

θ θθ θ

θ θ θ θθ θ

θ θ



55. 2 2tan 1 cos 1θ θ 2 2 2 2

22 2 2

2

2 2 2

2 2 2

2

2

tan cos tan cos 1

sincos tan cos 1

cos

sin tan cos 1

sin cos tan 1

1 tan 1

tan 2

θ θ θ θθ θ θ θθ

θ θ θ

θ θ θθ

θ

56. 2 2(cot 1)(sin 1)θ 2 2 2 2

22 2 2

2

2 2 2

2

2

cot sin cot sin 1

cossin cot sin 1

sin

cos sin cot 1

1 cot 1

cot 2

θ θ θ θθ θ θ θθθ θ θ

θθ

57. 2 2

2 2 2 2

2 2 2 2

(cos sin ) (cos sin )

cos 2cos sin sin cos 2cos sin sin

cos sin cos sin

1 1 2

θ θ θ θθ θ θ θ θ θ θ θθ θ θ θ

58. 2 2(3cos 4sin ) (4cos 3sin )θ θ θ θ 2 2

2 2

2 2 2 2

2 2 2 2

9cos 24cos sin 16sin

16cos 24cos sin 9sin

9cos 9sin 16sin 16cos

9(cos sin ) 16(sin cos )

9(1) 16(1)

25

θ θ θ θθ θ θ θθ θ θ θθ θ θ θ

59. 2 2

2

cos sin

1 tan

x x

x

2 2 2 2

2 2 2

2 2

2 2 2 2

2

2 2 22

2 2

cos sin cos sin

sin cos sin1

cos cos

cos sin cos sin

1 cos

cos sin coscos

1 cos sin

x x x x

x x x

x x

x x x x

x

x x xx

x x



60. sin cos cos sin

sin cos

x x x x

x x

2 2

2 2

(sin cos )cos (cos sin )sin

sin cos sin cos

sin cos cos cos sin sin

sin cos

cos sin

sin cos1

sin cos1 1

sin coscsc sec

sec csc

x x x x x x

x x x x

x x x x x x

x x

x x

x x

x x

x xx x

x x

61. Conjecture: left side is equal to cos x 2 2(sec tan )(sec tan ) sec tan

sec sec1

seccos

x x x x x x

x x

xx

62. Conjecture: left side is equal to sin x

2 2 22

2 2 2 2

2 2

2 2

1 1sec csc cos sinsincos

1 1sec csc cos sincos sin

sin

sin cossin

1sin

x x x xxxx x x x

x xx

x xx

x

63. Conjecture: left side is equal to 2sin x

cossin

cos cot sin cos cot sin

cot cot cotcos cot sin

cot

cos sinsin

cossin sin

2sin

xx

x x x x x x

x x xx x x

x

x xx

xx x

x



64. Conjecture: left side is equal to cos 1x cos tan tan 2cos 2 cos tan 2cos tan 2

tan 2 tan 2cos tan 2cos tan 2

tan 2 tan 2cos (tan 2)

1tan 2

cos 1

x x x x x x x x

x xx x x x

x xx x

xx

65. Conjecture: left side is equal to 2sec x 2 2 2 21 1 sec tan sec tan

sec tan sec tan sec tan sec tan(sec tan )(sec tan ) (sec tan )(sec tan )

sec tan sec tansec tan sec tan

2sec

x x x x

x x x x x x x xx x x x x x x x

x x x xx x x x

x

66. Conjecture: left side is equal to 2csc x

2 2

2 2

1 cos sin 1 cos 1 cos sin sin

sin 1 cos sin 1 cos 1 cos sin

1 2cos cos sin

sin 1 cos sin 1 cos

1 2cos cos sin

sin 1 cos

1 2cos 1

sin 1 cos

2 2cos

sin 1 cos

2(1 cos )

sin 1 cos

2

sin

x x x x x x

x x x x x x

x x x

x x x x

x x x

x x

x

x x

x

x x

x

x x

x

2csc x



67.

sin costan cot cos sin

1cscsin

x xx x x x

xx

2

2

2 2

sin cos sin

cos sin 1

sin sin cos

cos sin

1 cos cos

cos 1

1 cos cos

cos cos1

cos

x x x

x x

x x x

x x

x x

x

x x

x x

x

68.

1 1sec csc sin coscos sin

sin1 tan sin cos1cos

x x x xx xxx x xx

2

2

sin cos sin cos

cos sinsin cos

sin coscos

sin cos

sin cos sinsin cos

sin cos sin

1

sin

x x x x

x xx x

x xx

x x

x x xx x

x x x

x

69. cos cos cos sin 1 sin

tan1 sin 1 sin cos cos 1 sin

x x x x xx

x x x x x

2 2

2 2

2 2

cos sin sin

1 sin cos 1 sin cos

cos sin sin

1 sin cos

sin cos sin

1 sin cos

sin 1

1 sin cos

1

cos

x x x

x x x x

x x x

x x

x x x

x x

x

x x

x



70. 1 1 sin

cotsin cos sin cos cos

xx

x x x x x

2

2

1 cos cos

sin cos sin cos

1 cos

sin cos

sin

sin cossin

costan

1

cot

x x

x x x x

x

x x

x

x xx

xx

x

71. 1 cos 1 1 cos cos 1 cos

1 cos 1 cos 1 cos 1 cos 1 cos 1 cos

x x x x

x x x x x x

2

2 2

2

2

2

2

2

2 2

2 2

2 2

2

1 cos cos cos

1 cos 1 cos

1 cos cos cos

1 cos

1 cos

sin

1 cos

sin sin

csc cot

csc csc 1

2csc 1

x x x

x x

x x x

x

x

x

x

x x

x x

x x

x

72. (sec csc )(sin cos ) 2 cot sec sin sec cos csc sin csc cos 2 cotx x x x x x x x x x x x x x

sec sin sec cos csc sin csc cos 2 cot

tan 1 1 cot 2 cot

tan

x x x x x x x x x

x x x

x



73. 1 1

1csc sin sinsin

x x xx

2

2

2

2

11

sinsin

1

1 sin

sin sin1

1 sin

sin1

cos

sinsin

cos1 sin

cos cossec tan

xx

x

x x

x

x

x

xx

xx

x xx x

74. 1 sin 1 sin (1 sin )(1 sin ) (1 sin )(1 sin )

1 sin 1 sin (1 sin )(1 sin ) (1 sin )(1 sin )

x x x x x x

x x x x x x

2 2

2 2

2

2

(1 sin )(1 sin ) (1 sin )(1 sin )

(1 sin )(1 sin ) (1 sin )(1 sin )

1 2sin sin 1 2sin sin

1 sin 1 sin4sin

1 sin4sin

cos4 sin

cos cos4sec tan

x x x x

x x x x

x x x x

x xx

xx

xx

x xx x

75. – 78. Answers may vary.



79.

1

sec (sin cos ) 1 (sin cos ) 1cossin cos

1cos costan 1 1

tan

x x x x xxx x

x xx

x

80.

sin( )cos tan( ) cos

cos( )

sincos sin

cos

xx x x

x

xx x

x

81.

The graphs do not coincide. Values for x may vary.

82.


83.


84.


85.

2 2

sin sin 1csc

sin1 cos sin

x xx

xx x

86.

2 2sin sin cos sin (1 cos )

2 3sin sin sin

x x x x x

x x x

87.


88. makes sense

89. makes sense

90. makes sense

91. does not make sense; Explanations will vary. Sample explanation: The most efficient way to simplify the identity is to multiply out the numerator and then use a Pythagorean identity.



92. 3 3sin cossin cos

2 2(sin cos )(sin sin cos cos )

sin cos2 2sin cos sin cos

1 sin cos

x x

x x

x x x x x x

x x

x x x x

x x

93. sin cos 1sin cos 1

sin cos 1 sin cos 1

sin cos 1 sin cos 12 2sin 2cos sin cos 1

2 2sin 2sin cos 12 2sin cos 2cos sin 1

2 2sin 2sin (1 sin ) 1

1 2cos sin 12 2sin 2sin sin2cos sin

x x

x xx x x x

x x x x

x x x x

x x x

x x x x

x x x

x x

x x xx x

22sin 2sin

2sin cos

2sin (sin 1)

cos

sin coscos sin 1

sin 1 sin 1cos (sin 1)

2sin 1cos (sin 1)2 2cos 1 cos

cos (sin 1)2cos

sin 1

cos

x xx x

x x

x

x xx x

x xx x

xx x

x xx x

xx

x

94.

-

11ln | cos | ln | cos | ln| cos |

1ln ln | sec |

cos

x xx

xx

95. 2 2tan sec

2 2

2 2

2 2

ln

tan sec

( tan sec )

(sec tan )

1

x xe

x x

x x

x x

96. Answers may vary.


98. 3

cos302

1sin 30

21

cos602

3sin 60

2

cos90 0

sin 90 1

99. a. No, they are not equal.

cos(30 60 ) cos30 cos60

3 1cos90

2 2

1 30

2

b. Yes, they are equal.

cos(30 60 ) cos30 cos60 sin 30 sin 60

3 1 1 3cos90

2 2 2 2

3 30

4 40 0


sin(30 60 ) sin 30 sin 60

1 3sin 90

2 2

1 31

2

Section 6.2 Sum and Difference Formulas



sin(30 60 ) sin 30 cos60 cos30 sin 60

1 1 3 3sin 90

2 2 2 2

1 31

4 41 1

Section 6.2


1. cos30 cos(90 60 )

cos90 cos60 sin 90 sin 60

1 30 1

2 2

30

2

3

2

2. cos70 cos40 sin 70 sin 40

cos(70 40 )

cos30

3

2

3. cos( ) cos cos sin sincos cos cos cos

cos cos sin sin

cos cos cos cos

1 1 tan tan

1 tan tan

α β α β α βα β α β

α β α βα β α β

α βα β


4. 5sin sin12 6 4

sin cos cos sin6 4 6 4

1 2 3 2

2 2 2 2

2 6

4 4

2 6

4

π π π

π π π π

5. a. 4sin5

y

rα

Find x: 2 2 2

2 2 24 5

2 16 25

2 9

x y r

x

x

x

Because α is in Quadrant II, x is negative.

9 3x 3 3

cos5 5

x

rα

b. 1sin2

y

rβ

Find x: 2 2 2

2 2 21 2

2 1 4

2 3

x y r

x

x

x

Because β is in Quadrant I, x is positive.

3x

3cos

2

x

rβ

c. cos( ) cos cos sin sin

3 3 4 1

5 2 5 2

3 3 4

10 10

3 3 4

10

α β α β α β

d. sin( ) sin cos cos sin

4 3 3 1

5 2 5 2

4 3 3

10 10

4 3 3

10

α β α β α β

6. a. The graph appears to be the sine curve, sin .y x

It cycles through intercept, maximum, intercept, minimum and back to intercept. Thus, siny x also describes the graph.



b. 3 3 3cos cos cos sin sin2 2 2

x x xπ π π

cos 0 sin ( 1)

sin

x x

x

This verifies our observation that

3cos

2y x

π and siny x describe the

same graph.

7. tan tan

tan( )1 tan tan

tan 0

1 tan 0tan

1tan

xx

xx

xx

x

πππ


1. cos cos sin sinx y x y

2. cos cos sin sinx y x y

3. sin cos cos sinC D C D

4. sin cos cos sinC D C D

5. tan tan

1 tan tan

θ φθ φ

6. tan tan

1 tan tan

θ φθ φ

7. false

8. false

Exercise Set 6.2

1. cos(45 30 ) cos45 cos30 sin 45 sin 30

2 3 2 1

2 2 2 2

6 2

4 4

6 2

4

2. o ocos(120 45 )o o o ocos120 cos45 sin120 sin 45

1 2 3 2

2 2 2 2

2 6

4 4

2 6

4

3. 3 3 3

cos cos cos sin sin4 6 4 6 4 6

2 3 2 1

2 2 2 2

6 2

4 4

2 6

4

π π π π π π

4. 2 2 2

cos cos cos sin sin3 6 3 6 3 6

1 3 3 1

2 2 2 2

3 3

4 40

π π π π π π

5. a. cos50 cos20 sin50 sin 20

cos cos sin sin

Thus, 50 and 20 .

α β α βα β

b. cos50 cos20 sin50 sin 20

cos(50 20 )

cos30

c. 3

cos302

6. a. o o o ocos50 cos5 sin50 sin5

cos cos sin sinα β α β

Thus, o o50 and 5α β

b. o o o ocos50 cos5 sin50 sin5 o ocos(50 5 )

ocos 45



c. 2ocos45

2

7. a. 5 5

cos cos sin sin12 12 12 12

cos cos sin sin

5Thus, and .

12 12

π π π π

α β α βπ πα β

b. 5 5


5cos

12 12

4cos

12

cos3

π π π π

π π

π

π

c. 1

cos3 2

π

8. a. 5 5


π π π π

cos cos sin sin

5 and

18

α β α βπ πα β

b. 5 5


π π π π

5cos

18 9

3cos

18

cos6

π π

π

π

c. 3

cos6 2

π

9. cos( ) cos cos sin sin

cos sin cos sin

cos cos sin sin

cos sin cos sin

1 cot tan 1

tan cot



β αα β


sin sin sin sin

cos cos sin sin

sin sin sin sin

cos cos1

sin sin

cot cot 1


α β α βα β α βα βα βα β

11. cos cos cos sin sin4 4 4

2 2cos sin

2 2

2(cos sin )

2

x x x

x x

x x

π π π

12. 5 5 5

cos cos cos sin sin4 4 4

2 2cos sin

2 2

2(cos sin )

2

x x x

x x

x x

π π π

13. sin(45 30 ) sin 45 cos30 cos 45 sin 30

2 3 2 1

2 2 2 2

6 2

4 4

6 2

4

14. o osin(60 45 )o o o osin 60 cos 45 cos60 sin 45

3 2 1 2

2 2 2 2

6 2

4 4

6 2

4



15. sin(105 ) sin(60 45 )

sin 60 cos 45 cos60 sin 45

3 2 1 2

2 2 2 2

6 2

4 4

6 2

4

16. o o osin 75 sin(30 45 )o o o osin 30 cos 45 cos30 sin 45

1 2 3 2

2 2 2 2

2 6

4 4

2 6

4

17. cos(135 30 ) cos135 cos30 sin135 sin 30

cos 90 45 cos30 sin 90 45 sin 30

cos90 cos 45 sin 90 sin 45 cos30 sin 90 cos45 cos90 sin 45 sin 30

2 2 3 2 2 10 1 1 0

2 2 2 2 2 2

2 3 2 1

2 2 2 2

6 2

4 4

6 2

4

18. cos(240 45 ) cos240 cos 45 sin 240 sin 45

cos 180 60 cos 45 sin 180 60 sin 45

cos180 cos60 sin180 sin 60 cos45 sin180 cos60 cos180 sin 60 sin 45

1 3 2 1 3 21 0 0 ( 1)

2 2 2 2 2 2

1 2 3 2

2 2 2 2

2 6

4 4

6 2

4



19. cos75 cos(45 30 )

cos45 cos30 sin 45 sin 30

2 3 2 1

2 2 2 2

6 2

4 4

6 2

4

20. cos105 cos(45 60 )

cos 45 cos60 sin 45 sin 60

2 1 2 3

2 2 2 2

2 6

4 4

2 6

4

21. tan tan

6 4tan6 4 1 tan tan

6 4

π ππ π

π π

31

33

1 13

3 3

3 33 3

3 3

3 3

3 3

3 3 3 3

3 3 3 3

9 6 3 3

9 3

12 6 3

6

2 3

22. tan tan

3 4tan3 4 1 tan tan

3 4

π ππ π

π π

3 1

1 3 1

1 3

1 3

1 3 1 3

1 3 1 3

1 2 3 3

1 3

4 2 3

2

2 3

23.

4tan tan4 3 4tan

43 4 1 tan tan3 4

π ππ π

π π

3 1

1 3 1

1 3

1 3

1 3 1 3

1 3 1 3

1 2 3 3

1 3

4 2 3

2

2 3



24.

5tan tan5 3 4tan

53 4 1 tan tan3 4

π ππ π

π π

5tan tan

3 45

1 tan tan3 4

3 1

1 3 1

1 3

1 3

1 3 1 3

1 3 1 3

1 2 3 3

1 3

4 2 3

2

2 3

π π

π π

25. sin 25 cos5 cos25 sin5 sin(25 5 )

sin 30

1

2

26. o o o osin 40 cos 20 cos 40 sin 20 o osin(40 20 )osin 60

3

2

27. tan10 tan 35

tan(10 35 )1 tan10 tan 35

tan 45

1

28. o otan50 tan 20 o otan(50 20 )o o1 tan50 tan 20

otan 30

3

3

29. 5 5 5

sin cos cos sin sin12 4 12 4 12 4

2sin

12

sin6

1

2

π π π π π π

π

π

30. 7 7 7

sin cos cos sin sin12 12 12 12 12 12

6sin

12

sin2

1

π π π π π π

π

π

31. 5 30

5 30

tan tantan

5 301 tan tan

5tan tan

30 6

3

3

π π

π ππ π

π π

32.

4tan tan 45 5 tan

4 5 51 tan tan5 5

5tan

5tan

0

π ππ π

π π

π

π

33. sin sin cos cos sin2 2 2

sin 0 cos 1

cos

x x x

x x

x

π π π

34. 3 3 3

sin sin cos cos sin2 2 2

sin 0 cos ( 1)

cos

x x x

x x

x

π π π

35. cos cos cos sin sin2 2 2

cos 0 sin 1

sin

x x x

x x

x

π π π




1 cos 0 sin

cos

x x x

x x

x

π π π

37. tan 2 tan

tan(2 )1 tan 2 tan

0 tan

1 0 tantan

xx

xx

xx

πππ

38. tan tan

tan( )1 tan tan

0 tan

1 0 tantan

xx

xx

xx

πππ

39. sin( ) sin( )

sin cos cos sin

sin cos cos sin

2sin cos

α β α βα β α βα β α β

α β

40. cos( ) cos( )

cos cos sin sin

cos cos sin sin

2cos cos

α β α βα β α βα β α β

α β

41. sin( ) sin cos cos sin

cos cos cos cos

sin cos cos sin

cos cos cos cos

tan 1 1 tan

tan tan


α β α βα β α βα βα β

42. sin( ) sin cos cos sincos cos cos cos

sin cos cos sin

cos cos cos cos

tan tan


α β α βα β α βα β

43. 4

4

sin coscos coscos sincos cos

sin coscos

cos sincos

tan tantan

4 1 tan tan

tan 1

1 tan

sin cos cos

cos cos sinsin cos

cos sincos sin

cos sin

π

π

θ θθ θθ θθ θθ θ

θθ θ

θ

θπθθ

θθ

θ θ θθ θ θ

θ θθ θθ θθ θ

44. tan tan4tan

4 1 tan tan4

1 tan

1 1 tan1 tan

1 tancos sin

cos coscos sin

cos coscos sin

coscos sin

coscos sin cos sin

cos coscos sin cos

cos cos sincos sin

cos sin

π θπ θ π θ

θθ

θθ

θ θθ θθ θθ θθ θ

θθ θ

θθ θ θ θ

θ θθ θ θ

θ θ θθ θθ θ



45.

2 2 2 2

2 2 2 2

2 2 2

2 2 2

2 2

cos( ) cos( )

(cos cos sin sin )

(cos cos sin sin )

cos cos sin sin

1 sin cos sin 1 cos

cos sin cos

sin sin cos

cos sin



α β α β

β α βα α ββ α

46. sin sin(sin cos cos sin )

(sin cos cos sin )

2 2 2 2sin cos cos sin

2 2(1 cos )cos

2 2cos (1 cos )

2 2 2cos cos cos

2 2 2cos cos cos

2 2cos cos

a β α βα β α β

α β α β

α β α β

α β

α β

β α β

α α β

β α

47.

1cos cos

1cos cos

sin cos cos sincos cos


sin cos cos sincos cos cos cos

sin cosco

sin( )

sin( )

sin cos cos sin

sin cos cos sin

sin cos cos sin

sin cos cos sinα β

α βα β α β

α βα β α β

α βα β α βα β α βα β

α βα β



cos sins cos cos cos

tan 1 1 tan

tan 1 1 tan

tan tan

tan tan

α βα β α β

α βα βα βα β

48.

cos( )

cos

cos cos sin sin

cos cos sin sin

1

cos cos sin sin cos cos1cos cos sin sin

cos cos

cos cos sin sin

cos cos cos coscos cos sin sin

cos cos cos cos

1 tan tan

1 tan tan

α βα βα β α βα β α β


α βα β α βα β α βα β α βα β α β

α βα β

49. cos( ) cos

cos cos sin sin cos

cos cos cos sin sin

cos (cos 1) sin sin

cos 1 sincos sin

x h x

hx h x h x

hx h x x h

hx h x h

hh h

x xh h

50. sin( ) sin

sin cosh cos sinh sin

sin cosh sin cos sinh

sin (cosh 1) cos sinh

cosh 1 sinhsin cos

x h x

hx x x

hx x x

hx x

h

x xh h

51. sin 2 sin( )

sin cos cos sin

2sin cos

α α αα α α α

α α

52. cos2 cos( )α α α

cos cos sin sin

2 2cos sin

α α α α

α α



53.

2

tan 2 tan( )

tan tan

1 tan tan2 tan

1 tan

α α αα α

α αα

α

54. tan tan4 4

tan tan tan tan4 4

1 tan tan 1 tan tan4 4

1 tan 1 tan

1 tan 1 tan(1 tan )(1 tan ) (1 tan )(1 tan )

(1 tan )(1 tan ) (1 tan )(1 tan )2 21 2 tan tan (1 2 tan tan )

21 tan

π πα α

π πα α

π πα α

α αα αα α α αα α α α

α α α αα

4 tan21 tan

2(2 tan )21 tan

tan tan2

1 tan tan2 tan( )

2 tan 2

αα

αα

α αα α

α αα

55.

1cos cos

1cos cos


cos cos sin sincos cos

sin cos cos sincos cos cos cos

c

sin( )tan( )

cos( )

sin cos cos sin

cos cos sin sin

sin cos cos sin

cos cos sin sinα β

α βα β α β

α βα β α β

α βα β α βα β α β

α βα βα β



os cos sin sincos cos cos cos

tan tan

1 tan tan


α βα β

56. tan( ) tan( ( ))

tan tan( )

1 tan tan( )

tan tan

1 tan tan

α β α βα β

α βα β

α β

57. 3

sin5

y

rα

2 2 2

2 2 2

2

2

3 5

9 25

16

x y r

x

x

x

Because α lies in quadrant I, x is positive. x = 4

3545

4Thus, cos , and

5

sin 3tan .

cos 4

x

rα

ααα

5sin

13

y

rβ

2 2 2

2 2 2

2

2

5 13

25 169

144

x y r

x

x

x

Because β lies in quadrant II, x is negative. x = –12

5131213

12 12Thus, cos , and

13 13

sin 5tan .

cos 12

x

rβ

βββ

a. cos( ) cos cos sin sin

4 12 3 5 63

5 13 5 13 65

α β α β α β

b. sin( ) sin cos cos sin

3 12 4 5 16

5 13 5 13 65

α β α β α β

c.

3 5 44 12 12

633 5484 12

tan tantan( )

1 tan tan

16

631

α βα βα β



58. 4

sin5

y

rα

2 2 2

2 2 24 5

2 16 25

2 9

x y r

x

x

x

Because α lies in quadrant I, x is positive. x = 3

3cos

54

sin 45tan3cos 35

x

rα

ααα

7sin

25

y

rβ

2 2 2

2 2 27 25

2 49 625

2 576

x y r

x

x

x

Because β lies in quadrant II, x is negative. x = –24

24cos

257

sin 725tan24cos 24

25

x

rβ

βββ

a. cos( ) cos cos sin sinα β α β α β

3 24 4 7 4

5 25 5 25 5

b. sin sin cos cos sinα β α β α β

4 24 3 4 3

5 25 5 25 5

c. tan tan

tan( )1 tan tan

α βα βα β

4 7 2533 24 24

254 7 41

183 24

59. 3 3

tan4 4

y

xα

2 2 2

2 2 2

2

2

( 4) 3

16 9

25

x y r

r

r

r

Because r is a distance, it is positive. r = 5

4 4Thus, cos , and

5 53

sin .5

x

ry

r

α

α

1cos

3

x

rβ

2 2 2

2 2 2

2

2

1 3

1 9

8

x y r

y

y

y

Because β lies in quadrant I, y is positive. 8 2 2y

2 2313

2 2Thus, sin , and

3

sintan 2 2.

cos

y

rβ

βββ


4 1 3 2 2

5 3 5 3

4 6 2

15 15

4 6 2

15

4 6 2

15

α β α β α β


3 1 4 2 2

5 3 5 3

3 8 2

15 15

3 8 2

15

α β α β α β



c.

34

34

3 8 24

4 6 24

tan tantan( )

1 tan tan

2 2

1 2 2

4 6 23 8 2

4 6 2 4 6 2

108 50 2

56

54 25 2

28

α βα βα β

60. 4 4

tan3 3

y

xα

2 2 2

2 2 2( 3) 4

29 16

225

x y r

r

r

r

Because r is a distance, it is positive. r = 5

3cos

54

sin5

x

ry

r

α

α

2cos

3

x

rβ

2 2 2

2 2 22 3

24 9

2 5

x y r

y

y

y

Because β lies in quadrant I, y is positive. 5

sin3

5sin 53tan

2cos 23

y

rβ

βββ

5y


3 2 4 5 6 4 5

5 3 5 3 15

b. sin sin cos cos sinα β α β α β 4 2 3 5 8 3 5

5 3 5 3 15

c. tan tantan( )

1 tan tan

4 5

3 24 5

13 2

8 3 5

66 4 5

6

8 3 5

6 4 5

8 3 5 6 4 5

6 4 5 6 4 5

108 50 5

44

54 25 5

22

α βα βα β

61. 8

cos17

x

rα

2 2 2

2 2 2

2

2

8 17

64 289

225

x y r

y

y

y

Because α lies in quadrant IV, y is negative. y = –15

15178

17

15 15Thus, sin , and

17 17

sin 15tan .

cos 8

y

rα

ααα

1 1sin

2 2

y

rβ



2 2 2

2 2 2

2

2

( 1) 2

1 4

3

x y r

x

x

x

Because β lies in quadrant III, x is negative. 3x

12

32

3 3Thus, cos , and

2 2

sin 1 3tan .

cos 33

x

rβ

βββ


8 3 15 1

17 2 17 2

8 3 15

34

8 3 15

34

α β α β α β


15 3 8 1

17 2 17 2

15 3 8

34

α β α β α β

c. tan tantan1 tan tan

15 3

8 3

15 31

8 3

45 8 3

2424 15 3

24

45 8 3 24 15 3

24 15 3 24 15 3

1440 867 3

99

489 289 3

33

α βα βα β

62. 1

cos2

x

rα

2 2 2

2 2 21 2

21 4

2 3

x y r

y

y

y

Because α lies in quadrant IV, y is negative. 3y

3sin

2

3sin 2tan 3

1cos2

y

rα

ααα

1 1

sin3 3

y

rβ

2 2 2

22 21 3

2 1 9

2 8

x y r

x

x

x

Because β lies in quadrant III, x is negative. 8 2 2x

2 2cos

31

sin 1 23tancos 42 2 2 2

3

x

rβ

βββ


1 2 2 3 1

2 3 2 3

2 6 3

6


3 2 2 1 1

2 3 2 3

2 6 1

6



c.

tan tantan( )

1 tan tan

23

42

1 34

4 3 2

44 6

4

4 3 2

4 6

4 3 2 4 6

4 6 4 6

16 3 4 18 4 2 12

10

16 3 12 2 4 2 2 3

10

18 3 16 2

10

8 2 9 3

5

α βα βα β

63. 3

tan4

y

xα

Because α lies in quadrant III, x and y are negative.

2 2 2

2 22

2

4 3

25

5

r x y

r

r

r

3 3sin

5 54 4

cos5 5

y

rx

r

α

α

1

cos4

x

rβ

Because β lies in quadrant IV, y is negative.

2 2 2

2 2 2

2

1 4

15

15

x y r

y

y

y

15 15sin

4 4

15tan 15

1

y

r

y

x

β

β


4 1 3 15

5 4 5 4

4 3 15

20 20

4 3 15

20


3 1 4 15

5 4 5 4

3 4 15

20 20

3 4 15

20

c. tan tan

tan( )1 tan tan

α βα βα β

315

43

1 154

3 4 15

4 44 3 15

4 4

3 4 15

4 3 15

3 4 15 4 3 15

4 3 15 4 3 15

12 9 15 16 15 180

16 135

192 25 15

119

192 25 15

119



64. 5

sin6

y

rα

Because α lies in quadrant II, x is negative. 2 2 2

2 2 2

2

5 6

11

11

x y r

x

x

x

11 11cos

6 6

5 5 11tan

1111

x

r

y

x

α

α

3

tan7

y

xβ

Because β lies in quadrant III, x and y are both negative.

2 2 2

2 2 2

2

7 3

58

58

r x y

r

r

r

3 3 58sin

5858

7 7 58cos

5858

y

r

x

r

β

β


11 7 58 5 3 58

6 58 6 58

7 638 15 58

348 348

7 638 15 58

348


sin cos cos sin

5 7 58 11 3 58

6 58 6 58

35 58 3 638

348 348

3 638 35 58

348

α β α β

c. tan tan

tan( )1 tan tan

α βα βα β

5 11 3

11 7

5 11 31

11 7

35 11 33

77 7777 15 11

77 77

35 11 33

77 15 11

33 35 11 77 15 11

77 15 11 77 15 11

2541 495 11 2695 11 5775

5929 2475

8316 3190 11

3454

22 378 145 11

22 157

378 145 11

157

65. a. The graph appears to be the sine curve,

y = sin x. It cycles through intercept, maximum, minimum and back to intercept. Thus, y = sin x also describes the graph.


0 cos ( 1) sin

sin

x x x

x x

x

π π π

This verifies our observation that y = sin (π - x) and y = sin x describe the same graph.

66. a. The graph appears to be the cosine curve, y = cos x. It cycles through maximum, intercept, minimum, intercept and back to maximum. Thus, y = cos x also describes the graph.

b. cos( 2 ) cos cos2 sin sin 2x x xπ π π

cos 1 sin 0

cos

x x

x

This verifies our observation that cos( 2 )y x π and y = cos x describe the

same graph.



67. a. The graph appears to be 2 times the cosine curve, y = 2 cos x. It cycles through maximum, intercept, minimum, intercept and back to maximum. Thus y = 2 cos x also describes the graph.

b. sin sin2 2

sin cos cos sin sin cos2 2 2

cos sin2

sin 0 cos 1 1 cos 0 sin

cos cos

2cos

x x

x x x

x

x x x x

x x

x

π π

π π π

π


sin sin2 2

y x xπ π and y = 2cos x

describe the same graph.

68. a. The graph appears to be 2 times the sine curve, y = 2 sin x. It cycles through intercept, maximum, intercept, minimum and back to intercept. Thus, y = 2 sin x also describes the graph.

b. cos cos2 2

x xπ π

cos cos sin sin2 2

cos cos sin sin2 2

2sin sin2

2sin 1

2sin

x x

x x x

x

x

x

π π

π π

π


cos cos2 2

x xπ π and y = 2 sin x

describe the same graph.

69.

cos cos sin sin

cos

cos

α β β α β β

α β βα

70.

sin cos cos sin

sin

sin

α β β α β β

α β βα

71.

sin sin

cos cos


sin cos cos sin sin cos cos sin

cos cos sin sin cos cos sin sin



cos sin cos sin

cos cos cos cos

2cos sin

2cos cos

sin

α β α β α β α βα β α β α β α β

α β α β α β α βα β α β α β α βα β α βα β α β

α βα β

cos

tan

βββ

72.

cos cos

sin sin






cos cos cos cos

cos sin cos sin

2cos cos

2cos sin

c



α β α βα β α βα βα β

ossin

cot

βββ

73. cos cos sin sin6 6 6 6

π π π πα α α α

cos6 6

cos6 6

cos3

1

2

π πα α

π πα α

π



74. sin cos cos sin3 3 3 3

π π π πα α α α

sin3 3

sin3 3

2sin

3

3

2

π πα α

π πα α

π

75. Conjecture: the left side is equal to cos3x . cos2 cos5 sin 2 sin5

cos(2 5 )

cos( 3 )

cos3

x x x x

x x

x

x

76. Conjecture: the left side is equal to sin 3x . sin5 cos 2 cos5 sin 2

sin(5 2 )

sin 3

x x x x

x x

x

77. Conjecture: the left side is equal to sin2

x.

5 5sin cos2 cos sin 2

2 25

sin 22

5 4sin

2 2

sin2

x xx x

xx

x x

x

78. Conjecture: the left side is equal to cos2

x.

5 5cos cos 2 sin sin 2

2 25

cos 22

5 4cos

2 2

cos2

x xx x

xx

x x

x

79. 3

tan2

y

xθ

2 2 2

2 3 2

2

2

2 3

4 9

13

x y r

r

r

r

Because r is a distance, it is positive.

13r 3

Thus, sin132

and cos13

y

r

x

r

θ

θ

13 cos( ) 13(cos cos sin sin )

2 313 cos sin

13 13

cos 2 sin 3

2cos 3sin

t t t

t t

t t

t t

θ θ θ

For the equation 13 cos( ),y t θ the amplitude is

13 13 , and the period is 2 2 .1

π π

80. a. 3sin 2 2sin(2 )

3sin 2 2(sin 2 cos cos 2 cos )

3sin 2 2(sin 2 ( 1) cos 2 0)

3sin 2 2sin 2

sin 2

t t

t t t

t t t

t t

t

ρ ππ π

b. No. The amplitude of p is 1.

81. – 87. Answers may vary.

88.

–2

2

3–

232

3 3 3cos cos cos sin sin

2 2 2

0 cos 1 sin

sin

x x x

x x

x

π π π



89.

–2

2

3–

232

tan tan

tan( )1 tan tan

0 tan

1 0 tantan

1tan

xx

xx

xx

x

πππ

90.

–2

2

3–

232


91.

–2

2

3–

232


92.

–2

2

3–

232

cos1.2 cos0.8 sin1.2 sin 0.8

cos(1.2 0.8) cos 2

x x x x

x

93.

–2

2

3–

232

sin1.2 cos0.8 cos1.2 sin 0.8

sin(1.2 0.8 )

sin 2

x x x x

x x

x



94. makes sense

95. makes sense

96. does not make sense; Explanations will vary. Sample explanation: The sum and difference formulas allow you to find exact values only for certain angles.

97. makes sense

98. sin( ) sin( ) sin( )cos cos cos cos cos cos

sin cos cos sin sin cos cos sin sin cos cos sin

cos cos cos cos cos cos

sin cos cos sin sin cos cos sin sin

cos cos cos cos cos cos cos cos

x y y z z x

x y y z z x

x y x y y z y z z x z x

x y y z z x

x y x y y z y z

x y y y y z y z

cos cos sincos cos cos cos

sin sin sin sin sin sin

cos cos cos cos cos cos

0

z x z x

z x z x

x y y z z x

x y y z z x

99. 11cos2

1

3

2

31sin5

4

3

5

x

y

r

x

y

r

1 31 1sin cos sin2 5

1 31 1sin cos cossin2 5

1 31 1 coscos sin sin2 5

3 4 1 3

2 5 2 5

4 3 3

10



100. 31sin5

3, 5, 4y r x

41cos5

4, 3, 5x y r

3 41 1sin sin cos5 5

3 4 3 41 1 1 1sin sin coscos cossin sin cos5 5 5 5

3 4 4 3

5 5 5 5

12 12

25 2524

25

101. 41tan3

3

4

5

51cos13

5

12

13

x

y

r

x

y

r

4 51 1cos tan cos3 13

4 51 1cos tan coscos3 13

4 51 1 sin tan sin cos3 13

3 5 4 12

5 13 5 1333

65



102. 3 5 11 -1cos sin

2 6 2 6

π π

3 11 1cos cos sin2 2

5cos

6 6

cos

1

π π

π

103. Let 1sin , where - . sin2 2

x xπ πα α α

Because x is positive, sin α is positive. Thus α is in quadrant I. Using a right triangle in quadrant I with sin α = x, the third side can be found using the Pythagorean Theorem.

2 2 21

2 21

21

21 2Thuscos 11

a x

a x

a x

xxα

Because y is positive, cos β is positive. Thus β is in quadrant I. Using a right triangle in quadrant I with cos β = y, the third side can be found using the Pythagorean Theorem.

2 2 21

2 21

21

21 2Thuscos 11

b y

b x

a y

yyα

1 1cos(sin cos ) cos( )

cos cos sin sin

2 21 1

2 21 1

x y

x y x y

y x x y

α βα β α β

104. 1 1tan sinx x

1 1

2 2 1 1

y x y y

x r

r x x y

1 1sin(tan sin )

1 1sin tan cossin

1 1cos tan sin sin

21 1

12 21 1

21

2 1

x y

x y

x y

yxy

x x

x y y

x

105. 1sin 21

1

1cos

21

1

x x

y x

r

y

x y

y y

r

1 1tan(sin cos )

1 1tansin tan cos1 11 tansin tan cos

21

2121

121

2 21 1

2 21 1

x y

x y

x y

yx

yx

yx

yx

xy y x

y x x y


Section 6.3 Double-Angle Power Reducing, and Half-Angle Formulas


107. 1

sin 302

3cos30

2

3sin 60

21

cos602


sin(2 30 ) 2sin 30

1sin 60 2

2

31

2


sin(2 30 ) 2sin 30 cos30

1 3sin 60 2

2 2

3 3

2 2


cos(2 30 ) 2cos30

3cos60 2

21

32

b. Yes, they are equal. 2 2

2 2

cos(2 30 ) cos 30 sin 30

3 1cos60

2 2

1 3 1

2 4 41 1

2 2

Section 6.3


1. 4

sin5

y

rθ

Because θ lies in quadrant II, x is negative. 2 2 2

2 2 2

2 2 2

4 5

5 4 9

9 3

x y r

x

x

x

Now we use values for x, y, and r to find the required values.

a. sin 2 2sin cos

4 3 242

5 5 25

θ θ θ

b. 2 2

2 2

cos2 cos sin

3 4 9 16

5 5 25 25

7

25

θ θ θ

c.

2

4 8 83 3 3

2 16 74 9 93

2 tantan 2

1 tan

2

11

8 9 24

3 7 7

θθθ

2. The given expression is the right side of the formula for cos 2θ with 15 .θ

2 2 3cos 15 sin 15 cos(2 15 ) cos302

3.

2

2 2

2

2

3

3

sin 3 sin(2 )

sin 2 cos cos 2 sin

2sin cos cos (2cos 1)sin

2sin cos 2sin cos sin

4sin cos sin

4sin (1 sin ) sin

4sin 4sin sin

3sin 4sin

θ θ θθ θ θ θθ θ θ θ θ

θ θ θ θ θθ θ θθ θ θθ θ θθ θ

By working with the left side and expressing it in a form identical to the right side, we have verified the identity.



4. 24 22

2

2

sin sin

1 cos 2

2

1 2cos2 cos 2

41 1 1

cos 2 cos 24 2 41 1 1 1 cos 2(2 )

cos 24 2 4 2

1 1 1 1cos 2 cos4

4 2 8 83 1 1

cos 2 cos 48 2 8

x x

x

x x

x x

xx

x x

x x

5. Because 105° lies in quadrant II, cos105 0.

32

210cos105 cos

2

1 cos 210

2

1

2

2 3

4

2 3

2

6.

2

2

2

2

sin 2 2sin cos

1 cos 2 1 1 2sin

2sin cos

2 2sin2sin cos

2 1 sin

2sin cos

2cossin

tancos

θ θ θθ θ

θ θθ

θ θθ

θ θθ

θ θθ

The right side simplifies to tan ,θ the expression on the left side. Thus, the identity is verified.

7. 1

cos1 1 1

cos sin sin

1cos

cos1cos sin cos sin

1cos

1 coscos sin

sec

sec csc csc

1 cos sin

cos 1 cossin

1 cos

tan2

α

α α α

αα

α α α α

αα

α α

αα α α

α αα α

αα

α

We worked with the right side and arrived at the left side. Thus, the identity is verified.


1. 2sin cosx x

2. 2sin A ; 22cos A ; 22sin A

3. 2

2 tan

1 tan

B

B

4. 1 cos 2α

5. 1 cos2α

6. 1 cos 2 y

7. 1 cos x

8. 1 cos y

9. 1 cosα ; 1 cosα ; 1 cosα

10. false

11. false

12. false

13. +

14.

15. +



Exercise Set 6.3

1. 3 4 24

sin 2 2sin cos 25 5 25

θ θ θ

2. 2 2cos2 cos sinθ θ θ 2 2

4 3 16 9 7

5 5 25 25 25

3.

2

3 34 2

2 93 164

327

16

2 tantan 2

1 tan

2

11

3 16 24

2 7 7

θθθ

Use this information to solve problems 4, 5, and 6.

7tan

24

y

xα

Because r is a distance it is positive. 2 2 2

2 2 2

2

2

24 7

576 49

625

25

7sin

2524

cos25

x y r

r

r

r

r

y

rx

r

α

α

4. 7 24 336

sin 2 2sin cos 225 25 625

α α α

5. 2 2

2 2

cos2 cos sin

24 7 576 49

25 25 625 625

527

625

α α α

6. 2 tantan 221 tan

7 14 14224 24 24

2 49 5277 11 576 57624

14 576 336

24 527 527

ααα

7. 15

sin17

y

rθ

Because θ lies in quadrant II, x is negative. 2 2 2

2 2 2

2 2 2

15 17

17 15 64

64 8

x y r

x

x

x


a. sin 2 2sin cos

15 8 2402

17 17 289

θ θ θ

b. 2 2

2 2

cos2 cos sin

8 15 64 225

17 17 289 289

161

289

θ θ θ

c.

2

15 15 158 4 4

2 225 16115 64 648

2 tantan 2

1 tan

2

11

15 64 240

4 161 161

θθθ

8. 12

sin13

y

rθ

Because θ lies in quadrant II, x is negative.

2

2 2 2

2 2

2

12 13

25

25 5

x y r

x

x

x




a. sin 2 2sin cos

12 52

13 13

120

169

θ θ θ

b. 2 2cos2 cos sinθ θ θ

2 25 12

13 13

25 144

169 169119

169

c. 2 tantan 221 tan12 2425 5

2 14412 11 255

241205

119 1195

θθθ

9. 24

cos25

x

rθ

Because θ lies in quadrant IV, y is negative. 2 2 2

2 2 2

2 2 2

24 25

25 24 49

49 7

x y r

y

y

y


a. sin 2 2sin cos

7 24 3362

25 25 625

θ θ θ

b. 2 2

2 2

cos2 cos sin

24 7

25 25

576 49 527

625 625 625

θ θ θ

c.

2

7 7 724 12 12

2 49 5277 576 57624

2 tantan 2

1 tan

2

11

7 576 336

12 527 527

θθθ

10. 40

cos41

x

rθ

Because θ lies in quadrant IV, y is negative. 2 2 2

2 2 2

2

40 41

81

81 9

x y r

y

y

y


a. sin 2 2sin cosθ θ θ

9 40 720

241 41 1681


240 9 1600 81

41 41 1681 1681

1519

1681

c. 2 tan

tan 221 tan

θθθ

9 9240 20

2 819 11 160040

99 1600 72020

1519 20 1519 15191600



11. 2

cot 21

x

yθ

Because r is a distance, it is positive. 2 2 2

2 2 2

2

( 2) ( 1)

5

5

r x y

r

r

r


a. sin 2 2sin cos

1 2 42

55 5

θ θ θ

b. 2 2

2 2

cos2 cos sin

2 1

5 5

4 1 3

5 5 5

θ θ θ

c.

2

12

2 1 31 4 42

2 tantan 2

1 tan

2 1 1

11

4 4(1)

3 3

θθθ

12. 3

cot 31

x

yθ

Because r is a distance, it is positive.

2 2 2

2 22

2

3 1

10

10

r x y

r

r

r



1 3 6 3

210 510 10


2 23 1

10 10

9 1 8 4

10 10 10 5

c. 2 tan

tan 221 tan

θθθ

1 223 3

2 11 11 93

22 9 33

8 3 8 49

13. 9 9

sin41 41

y

rθ

Because θ lies in quadrant III, x is negative.

2 2 2

22 2

2

9 41

1600

1600

40

x y r

x

x

x

x


a. sin 2 2sin cos

9 40 7202

41 41 1681

θ θ θ



b. 2 2

2 2

cos2 cos sin

40 9

41 41

1600 81

1681 16811519

1681

θ θ θ

c.

2

9 9 940 20 20

2 81 15199 1600 160040

2 tantan 2

1 tan

2

11

9 1600 720

20 1519 1519

θθθ

14. 2 2

sin3 3

y

rθ

Because θ lies in quadrant III, x is negative.

2 2 2

22 2

2

2 3

5

5

x y r

x

x

x



2 5 4 5

23 3 9


2 25 2

3 3

5 4 1

9 6 9

c. 2 tan

tan 221 tan

θθθ

2 42

5 52 4

2 11 5

5

4

4 5 55 4 51 4 15

15. The given expression is the right side of the formula for sin 2 with 15 .θ θ 2sin15 cos15 sin(2 15 )

1sin 30

2

16. The given expression is the right side of the formula for sin 2θ with θ = 22.5o.

o o o2sin 22.5 cos 22.5 sin(2 22.5 )

2osin 452

17. The given expression is the right side of the formula for cos2 with 75 .θ θ

2 2cos 75 sin 75 cos(2 75 )

3cos150

2

18. The given expression is the right side of the formula for cos 2θ with θ = 105o

2 o 2 o ocos 105 sin 105 cos(2 105 )

3ocos 2102

19. The given expression is the right side of the formula

for cos2 with .8

πθ θ

22cos 1 cos 28 8

2cos

4 2

π π

π


for cos 2θ with θ = .12

π

321 2sin cos 2 cos12 12 6 2

π π π


for tan 2 with .12

πθ θ

122

12

2 tan 3tan 2 tan

12 6 31 tan

π

ππ π




for tan 2θ with θ = .8

π

2 tan8 tan 2 tan 1

8 421 tan8

ππ π

π

23. 2 2

2 2

2 2

2

2

sincos

2 cos sincos cos

2sincos

cos sincos

2sincos

1cos

2

22 tan

1 tan

2sin cos

cos 12sin cos

sin 2

θθ

θ θθ θ

θθ

θ θθ

θθ

θ

θθ

θ θθθ θθ

24. cos22cot sin2 2 21 cot sin cos

2 2sin sin2cos

sin2 2sin cos

2sin2cos

sin12sin

22cos sin

sin 12cos sin

sin 2

θθ θ

θ θ θθ θ

θθ

θ θθ

θθ

θθ θ

θθ θ

θ

25. 2 2 2

2 2

(sin cos ) sin 2sin cos cos

sin cos 2sin cos

1 2sin cos

1 sin 2

θ θ θ θ θ θθ θ θ θ

θ θθ

26. 2(sin cos )2 2sin 2sin cos cos

2 2sin cos 2sin cos

1 2sin cos

1 sin 2

θ θ

θ θ θ θ

θ θ θ θθ θθ

27. 2 2 2 2

2

sin cos 2 sin cos sin

cos

x x x x x

x

28. 2cos 2 1 2sin

2 2cos cos2 21 sin sin

2cos2 2cos sin

2cos2 2cos sin2 2cos cos

21 tan

x x

x x

x x

x

x x

x

x x

x x

x

29. 2 2

2 2

2 2

2

sin 2 2sin cos

1 cos 2 1 cos sin

2sin cos

1 cos sin2sin cos

sin sin2sin cos

2sincos

sincot

x x x

x x x

x x

x xx x

xx x

xx

xx

30. 2 21 cos 2 1 cos sin

sin 2 2sin cos2 21 sin cos

2sin cos2 2cos cos

2sin cos22cos

2sin coscos

sincot

x x x

x x x

x x

x x

x x

x x

x

x xx

xx



31. 22

sintan cos2 2cos 1

cos

2sin cos sin

cos cos2sin cos tan

sin 2 tan

tt t t

t

t t t

t tt t t

t t

32. cos 2cos cos 2 1 2sinsin2cos 2cos sin

sin sincot 2cos sin

2cos sin cot

sin 2 cot

tt t t

t

t t t

t tt t t

t t t

t t

33.

2 23 3

sin 4 sin(2 2 )

sin 2 cos 2 cos 2 sin 2

cos2 (sin 2 sin 2 )

cos2 2sin 2

cos sin 2 2sin cos

4sin cos 4sin cos

t t t

t t t t

t t t

t t

t t t t

t t t t

34. cos4 cos 2(2 )22cos 2 1

2 22(2cos 1) 1

4 22(4cos 4cos 1) 1

4 28cos 8cos 2 1

4 28cos 8cos 1

t t

t

t

t t

t t

t t

35. 46sin x 2

1 cos26

2

21 2cos2 cos 26

4

26 12cos 2 6cos 2

43 3 23cos2 cos 24 23 3 1 cos 4

3cos24 2 2

3 3 1 cos 43cos2

4 2 2 2

3 3 33cos2 cos 4

4 4 49 3

3cos2 cos 44 4

x

x x

x x

x x

xx

xx

x x

x x

36.

1 cos 2 1 cos22 210cos cos 102 2

210(1 2cos 2 cos 2 )

4210 20cos 2 10cos 2

4210 20cos2 10cos 2

4 4 45 5

5cos 2 1 cos 42 45 5 5

5cos 2 cos 42 4 415 5

5cos 2 cos44 4

x xx x

x x

x x

x x

x x

x x

x x



37. 2 2

2

2

1 cos2 1 cos 2sin cos

2 2

1 cos 2

41 1

cos 24 41 1 1 cos(2 2 )

4 4 2

1 1(1 cos 4 )

4 81 1 1

cos 44 8 81 1

cos 48 8

x xx x

x

x

x

x

x

x

38. 1 cos 2 1 cos 22 28sin cos 8

2 228(1 cos 2 )

428 8(cos 2 )

4 41 cos 2 2

2 22

2 1 cos 4

1 cos 4

x xx x

x

x

x

x

x

39. Because 15° lies in quadrant I, sin15 0.

32

30ºsin15 sin

2

11 cos30

2 2

2 3 2 3

4 2

40. Because 22.5° lies in quadrant I, cos 22.5o >0. o o45 1 cos 45ocos22.5 cos

2 2

21

22

2 2

4

2 2

2

41. Because 157.5° lies in quadrant II, cos157.5 0.

22

315 1 cos315cos157.5 cos

2 2

1 2 2

2 4

2 2

2

42. Because 105° lies in quadrant II, sin 105o > 0. o o210 1 cos210osin105 sin

2 2

31

2

2

2 3

4

2 3

2

43. Because 75° lies in quadrant I, tan 75 0.

3212

150 1 cos150tan 75 tan

2 sin150

12 3

44. Because 112.5° lies in quadrant II, tan 112.50 < 0. o225otan125 tan

2o1 cos225

osin 225

21

2

2

2

2 2

22

12

2 1



45. Because 7

8

π lies in quadrant II,

7tan 0.

8

π

7 74 4

74

22

1 cos7tan tan

8 2 sin

1 21

2 2

2

2 1

π π

ππ

46. Because 3

8

πlies in quadrant I, tan

3

8

π > 0.

33 4tan tan8 2

31 cos

43

sin4

21

2

2

22

12

2 1

ππ

π

π

47.

45

1 cossin

2 2

1 1

2 10

1 10

1010

θ θ

48. 1 cos

cos2 2

41

52

9

52

9

103

10

3 10

10

θ θ

49.

45

35

1 costan

2 sin1

1

3

θ θθ

Use this information to solve problems 50, 51, 52 and 54.

7tan

24

y

xα

Because r is a distance, it is positive. 2 2 2

2 2 2

2

24 7

625

25

r x y

r

r

r

7sin

2524

cos25

y

rx

r

α

α



50. 1 cos

sin2 2

241

252

1

252

1

501

5 2

2

10

α α

51. 242511 cos 49cos

2 2 2 50

7 7 2

105 2

α α

52. 1 cos

tan2 sin

241

257

251

257

251

7

α αα

53. 1 cos 1 cos

2sin cos 22 2 2 2

θ θ θ θ

4 45 51 12

2 2

1 92

10 101 3

210 10

6 3

10 5

54. 1 cos 1 cos

2sin cos 22 2 2 2

α α α α

24 241 1

25 2522 2

1 492

50 501 7

250 50

7

25

chapter 6 analytic trigonometry...2013/07/31 · chapter 6 analytic trigonometry ... a

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