6.5e: Exercises: Double Angle, Half Angle and Reductions Formulas
( \newcommand{\kernel}{\mathrm{null}\,}\)
A: Concepts.
Exercise 6.5e.A
1) Explain how to determine the reduction identities from the double-angle identity cos(2x)=cos2x−sin2x
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)=√1−cosx√1+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. 1−cosxsinx, sinx1+cosx, multiplying the top and bottom by √1−cosx 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 π2≤θ≤π | 12. cos(2θ)=1√2 and π≤θ≤3π2 |
- Answers to odd exercises.
-
5a. 3√732 5b. 3132 5c. 3√731 7a. √32 7b. −12 7c. −√3 9a. 120169, 9b. −119169, 9c. −120119
11. cosθ=−2√55,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)=9cos2θ−4 |
30. 3cos(2α)=2cos2(α) 31. cos(2x)−cosx=0 32. 4sin2x+sin(2x)secx−3=0 33. sin2x−1+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. 1−2sin2(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. (sin2x−1)2=cos(2x)+sin4x 52. (sint−cost)2=1−sin(2t) 53. sin(2x)=−2sin(−x)cos(−x) 54. cotx−tanx=2cot(2x) 55. 1+cos(2θ)sin(2θ)tan2θ=tanθ 56. 1+cos(2t)sin(2t)−cost=2cost2sint−1 |
57. sin(2x)=2tanx1+tan2x 58. cos(2α)=1−tan2α1+tan2α 59. tan(2x)=2sinxcosx2cos2x−1 60. sin(2θ)1+cos(2θ)=tan(θ) 61. cos(16x)=(cos2(4x)−sin2(4x)−sin(8x)) 62. sin(16x)=16sinxcosxcos(2x)cos(4x)cos(8x) |
- Answers to odd exercises.
-
51. (sin2x−1)2=sin4x−2sin2x+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+cos2θ−sin2θtan2θ=2sin(θ)cos(θ)2cos2θtan2θ=sin(θ)cos(θ)tan2θ=cot(θ)tan2θ=tanθ
57. 2tanx1+tan2x=2sinxcosx1+sin2xcos2x=2sinxcosxcos2x+sin2xcos2x=2sinxcosx⋅cos2x1=2sinxcosx=sin(2x)
59. 2sinxcosx2cos2x−1=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. 1−cos(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(cos2x−sin2x)
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 0≤θ≤2π
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.
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}
\bigstar