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6.5e: Exercises: Double Angle, Half Angle and Reductions Formulas

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A: Concepts.

Exercise 6.5e.A

1) Explain how to determine the reduction identities from the double-angle identity cos(2x)=cos2xsin2x

2) Explain how to determine the double-angle formula for tan(2x) using the double-angle formulas for cos(2x) and sin(2x).

3) We can determine the half-angle formula for tan(x2)=1cosx1+cosx by dividing the formula for sin(x2)  by cos(x2). Explain how to determine two formulas for tan(x2) that do not involve any square roots.

4) For the half-angle formula given in the previous exercise for tan(x2) , explain why dividing by 0 is not a concern. (Hint: examine the values of cosx necessary for the denominator to be 0.)

Answers to odd exercises.

1. Use the Pythagorean identities and isolate the squared term.

3. 1cosxsinx, sinx1+cosx, multiplying the top and bottom by 1cosx and 1+cosx, respectively.

B: Evaluate Double Angle Trigonometric Expressions

Exercise 6.5e.B

5. If sinx=18 and x is in quadrant I.

6. If sinx=23, and x is in quadrant II.

7. If cosx=12, and x is in quadrant III.

8. If cosx=512, and x is in quadrant I.

9. If tanx=125, and x is in quadrant I.

10. If tanx=8, and x is in quadrant IV.

 Find the values of the six trigonometric functions for angle θ if the conditions provided hold.

11. cos(2θ)=35 and π2θπ 12. cos(2θ)=12 and πθ3π2
Answers to odd exercises.

5a. 3732     5b. 3132     5c. 3731     7a.  32     7b.  12     7c.  3     9a. 120169,     9b.  119169,     9c. 120119     

11.  cosθ=255,sinθ=55,tanθ=12,cscθ=5,secθ=52,cotθ=2

C: Use Double Angle Formulas to Solve Equations 

Exercise 6.5e.C

  Use double angle formulas to solve exactly (where possible) on the interval [0,2π).

20. sin(2t)=cost

21. sin(2t)+sint=0

22. sin(2x)sinx=0

23. sin(2t)+3cos(t)=0

24. sin(4x)sin(2x)=0

25. sin(2x)sec2x=0

26. sin(2x)2csc2x=0 

27. cos(2t)=sint

28. cos(6x)cos(3x)=0

29. 9cos(2θ)=9cos2θ4

30. 3cos(2α)=2cos2(α)

31. cos(2x)cosx=0

32. 4sin2x+sin(2x)secx3=0

33. sin2x1+2cos(2x)cos2x=1

34.  8cos(2α)=8cos2(α)1

35. 6sin(2t)+9sin(t)=0

Answers to odd exercises.

21. 0,2π3,π,4π3     23. π2,3π2    25. 0,π     27. 3π2,π6,5π6     
29. 0.7297,2.4119,3.8713,5.5535     31. 0,2π3,4π3     33. No solution.     35. 0,π,2.4189,3.8643

D: Recognize patterns

Exercise 6.5e.D

  Simplify to one trigonometric expression. Do not evaluate.

41. 2sin(π4)2cos(π4)

42. 4sin(π8)cos(π8)

43. 6sin(5x)cos(5x)

44. 4sin(8x)cos(8x)

45. cos2(6x)sin2(6x)

46.  cos2(28)sin2(28)

47.  2cos2(37)1

48.  12sin2(17)

49. cos2(9x)sin2(9x)

50. cos2(37)sin2(37)

 

Answers to odd exercises.

41.  2sin(π2)     43. 3sin(10x)     45. cos(12x)       47. cos(74)   49. cos(18x)

E: Verify identities

Exercise 6.5e.E

  Prove the identity.

51. (sin2x1)2=cos(2x)+sin4x

52. (sintcost)2=1sin(2t)

53. sin(2x)=2sin(x)cos(x)

54. cotxtanx=2cot(2x)

55. 1+cos(2θ)sin(2θ)tan2θ=tanθ

56. 1+cos(2t)sin(2t)cost=2cost2sint1

57. sin(2x)=2tanx1+tan2x

58. cos(2α)=1tan2α1+tan2α

59. tan(2x)=2sinxcosx2cos2x1

60. sin(2θ)1+cos(2θ)=tan(θ)

61. cos(16x)=(cos2(4x)sin2(4x)sin(8x))
                           (cos2(4x)sin2(4x)+sin(8x))

62. sin(16x)=16sinxcosxcos(2x)cos(4x)cos(8x)

Answers to odd exercises.

51. (sin2x1)2=sin4x2sin2x+1=cos(2x)+sin4x

53. 2sin(x)cos(x)=2(sin(x)cos(x))=sin(2x)

55. sin(2θ)1+cos(2θ)tan2θ=2sin(θ)cos(θ)1+cos2θsin2θtan2θ=2sin(θ)cos(θ)2cos2θtan2θ=sin(θ)cos(θ)tan2θ=cot(θ)tan2θ=tanθ

57. 2tanx1+tan2x=2sinxcosx1+sin2xcos2x=2sinxcosxcos2x+sin2xcos2x=2sinxcosxcos2x1=2sinxcosx=sin(2x)

59. 2sinxcosx2cos2x1=sin(2x)cos(2x)=tan(2x)

61. (cos2(4x)sin2(4x)sin(8x))(cos2(4x)sin2(4x)+sin(8x))=(cos(8x)sin(8x))(cos(8x)+sin(8x))=cos2(8x)sin2(8x)=cos(16x)

F: Use Power Reduction Formulas

Exercise 6.5e.F

  Rewrite the expression with an exponent no higher than 1.

70. cos2(5x)

71. cos2(6x)

72. sin4(8x)

73. sin4(3x)

74. sin2(2x)

75. sin2xcos2x

76. tan2xsin2x

77. tan4x

78. cos2xsin(2x)

79. tan4xcos2x

80. tan2(x2)sinx

81. cos2(2x)sinx

82. tan2xsinx

83. cos4xsin2x

84. cos2xsin4x

  Algebraically find an equivalent function, in terms of only sinx and/or cosx

85. sin(4x) 86. cos(4x)
Answers to odd exercises.

71. 1+cos(12x)2     73. 3+cos(12x)4cos(6x)8     75. 1cos(4x)8     
77. 3+cos(4x)4cos(2x)3+cos(4x)+4cos(2x)     79. 3+cos(4x)4cos(2x)4(cos(2x)+1)     81. (1+cos(4x))sinx2
83. 2+cos(2x)2cos(4x)cos(6x)32      85. 4sinxcosx(cos2xsin2x)

G: Use Half Angle Formulas 

Exercise 6.5e.G

  Find the exact value using half-angle formulas.

91. sin(π8)

92. cos(11π12)

93. sin(11π12)

94. cos(7π8)

95. tan(5π12)

96. tan(3π12)

97. tan(3π8)

 

  Find the exact values of a) sin(x2) b) cos(x2) and c) tan(x2) when 0θ2π

101. If sinx=1213, and x is in quadrant III.

102. If tanx=43, and x is in quadrant IV.

103. If secx=4, and x is in quadrant II.

104. If cscx=7, and x is in quadrant II.

  Use the figure below to find the requested half angle trigonometric expressions.

CNX_Precalc_Figure_07_03_201.jpg

107. Find sin(α2), \cos \left (\dfrac{\alpha }{2} \right ), and \tan \left (\dfrac{\alpha }{2} \right ).

108. Find \sin \left (\dfrac{\theta }{2} \right ), \cos \left (\dfrac{\theta }{2} \right ), and \tan \left (\dfrac{\theta }{2} \right )\\[4pt].

Answers to odd exercises.

91.  \dfrac{\sqrt{2-\sqrt{2}}}{2}     93.  \dfrac{\sqrt{2-\sqrt{3}}}{2}     95.  2+\sqrt{3}     97.  -1-\sqrt{2}     

101a.  \dfrac{3\sqrt{13}}{13}     101b.  -\dfrac{2\sqrt{13}}{13}     101c.  -\dfrac{3}{2}     103a. \dfrac{\sqrt{10}}{4}     103b. \dfrac{\sqrt{6}}{4}     103c. \dfrac{\sqrt{15}}{3}

107. \dfrac{2\sqrt{13}}{13}, \dfrac{3\sqrt{13}}{13}, \dfrac{2}{3}

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6.5e: Exercises: Double Angle, Half Angle and Reductions Formulas is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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