6.5e: Exercises: Double Angle, Half Angle and Reductions Formulas
- Page ID
- 74636
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A: Concepts.
Exercise \(\PageIndex{A}\)
1) Explain how to determine the reduction identities from the double-angle identity \(\cos(2x)=\cos^2x-\sin^2x\)
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 \(\tan \left ( \dfrac{x}{2} \right )=\dfrac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}\) by dividing the formula for \(\sin \left ( \dfrac{x}{2} \right )\) by \(\cos \left ( \dfrac{x}{2} \right )\). Explain how to determine two formulas for \(\tan \left ( \dfrac{x}{2} \right )\) that do not involve any square roots.
4) For the half-angle formula given in the previous exercise for \(\tan \left ( \dfrac{x}{2} \right )\) , explain why dividing by \(0\) is not a concern. (Hint: examine the values of \(\cos x\) necessary for the denominator to be \(0\).)
- Answers to odd exercises.
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1. Use the Pythagorean identities and isolate the squared term.
3. \(\dfrac{1-\cos x}{\sin x}\), \(\dfrac{\sin x}{1+\cos x}\), multiplying the top and bottom by \(\sqrt{1-\cos x}\) and \(\sqrt{1+\cos x}\), respectively.
B: Evaluate Double Angle Trigonometric Expressions
Exercise \(\PageIndex{B}\)
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5. If \(\sin x =\dfrac{1}{8}\) and \(x\) is in quadrant \(\mathrm{I} \\[4pt]\). 6. If \(\sin x =\dfrac{2}{3}\), and \(x\) is in quadrant \(\mathrm{II} \\[4pt] \). 7. If \(\cos x =-\dfrac{1}{2}\), and \(x\) is in quadrant \(\mathrm{III} \). |
8. If \(\cos x = \dfrac{5}{12} \), and \(x\) is in quadrant \(\mathrm{I}\). 9. If \(\tan x = \dfrac{12}{5}\), and \(x\) is in quadrant \(\mathrm{I} \\[4pt] \). 10. If \(\tan x =-8\), and \(x\) is in quadrant \(\mathrm{IV} \\[4pt] \). |
\( \bigstar \) Find the values of the six trigonometric functions for angle \(\theta\) if the conditions provided hold.
| 11. \(\cos(2\theta )=\dfrac{3}{5}\) and \( \frac{\pi}{2} \leq \theta \leq \pi\) | 12. \(\cos(2\theta )=\dfrac{1}{\sqrt{2}}\) and \( \pi \leq \theta \leq \frac{3\pi}{2} \) |
- Answers to odd exercises.
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5a. \(\dfrac{3\sqrt{7}}{32}\) 5b. \(\dfrac{31}{32}\) 5c. \(\dfrac{3\sqrt{7}}{31}\) 7a. \(\dfrac{\sqrt{3}}{2}\) 7b. \(-\dfrac{1}{2}\) 7c. \(-\sqrt{3}\) 9a. \(\dfrac{120}{169},\) 9b. \( -\dfrac{119}{169}, \) 9c. \(-\dfrac{120}{119}\)
11. \(\cos \theta =-\dfrac{2\sqrt{5}}{5},\sin \theta =\dfrac{\sqrt{5}}{5},\tan \theta =-\dfrac{1}{2},\csc \theta =\sqrt{5},\sec \theta =-\dfrac{\sqrt{5}}{2},\cot \theta =-2\)
C: Use Double Angle Formulas to Solve Equations
Exercise \(\PageIndex{C}\)
\( \bigstar \) Use double angle formulas to solve exactly (where possible) on the interval \([0,2\pi )\).
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20. \(\sin(2t)=\cos t \\[2pt]\) 21. \( \sin(2t)+ \sin t=0 \\[2pt]\) 22. \(\sin(2x)-\sin x=0 \\[2pt] \) 23. \( \sin (2t) +3\cos (t) =0 \\[2pt]\) 24. \(\sin (4x)-\sin (2x)=0 \) |
25. \(\dfrac{\sin (2x)}{\sec ^2 x}=0 \\[4pt] \) 26. \(\dfrac{\sin (2x)}{2\csc ^2 x}=0 \\[4pt] \) 27. \(\cos(2t)=\sin t \\[2pt]\) 28. \(\cos(6x)-\cos(3x)=0 \) 29. \(9\cos(2\theta )=9\cos^2\theta -4 \\[2pt]\) |
30. \(3\cos (2\alpha )=2\cos ^{2} (\alpha ) \) 31. \(\cos(2x)-\cos x=0 \) 32. \(4\sin^2 x+\sin(2x)\sec x-3=0 \) 33. \(\sin^2 x-1+2\cos(2x)-\cos^2 x=1 \) 34. \(8\cos \left(2\alpha \right)=8\cos ^{2} \left(\alpha \right)-1\) 35. \(6\sin \left(2t\right)+9\sin \left(t\right)=0\) |
- Answers to odd exercises.
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21. \( 0, \dfrac{2\pi }{3}, \pi , \dfrac{4\pi }{3} \) 23. \( \dfrac{\pi }{2}, \dfrac{3\pi }{2}\) 25. \(0, \pi \) 27. \(\dfrac{3\pi }{2}, \dfrac{\pi }{6}, \dfrac{5\pi }{6}\)
29. \( 0.7297, \; 2.4119, \; 3.8713, \; 5.5535 \) 31. \(0, \dfrac{2\pi }{3}, \dfrac{4\pi }{3}\) 33. No solution. 35. \( 0, \; \pi, \; 2.4189, \; 3.8643 \)
D: Recognize patterns
Exercise \(\PageIndex{D}\)
\( \bigstar \) Simplify to one trigonometric expression. Do not evaluate.
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41. \(2\sin \left ( \dfrac{\pi }{4} \right )2\cos \left ( \dfrac{\pi }{4} \right )\\[4pt]\) 42. \(4\sin \left ( \dfrac{\pi }{8} \right )\cos \left ( \dfrac{\pi }{8} \right )\\[4pt]\) |
43. \(6\sin (5x)\cos (5x) \\[4pt]\) 44. \(4\sin (8x)\cos (8x) \\[4pt]\) 45. \(\cos ^{2} \left(6x\right)-\sin ^{2} (6x)\) |
46. \(\cos ^2(28^{\circ})-\sin ^2(28^{\circ}) \\[4pt]\) 47. \(2\cos ^2(37^{\circ})-1 \\[4pt]\) 48. \(1-2\sin ^2(17^{\circ})\) |
49. \(\cos ^2(9x)-\sin ^2(9x) \\[4pt]\) 50. \(\cos ^{2} \left(37{}^\circ \right)-\sin ^{2} (37{}^\circ ) \\[4pt]\)
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- Answers to odd exercises.
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41. \(2\sin \left ( \dfrac{\pi }{2} \right )\) 43. \(3\sin (10x)\) 45. \(\cos ( 12x )\) 47. \(\cos (74^{\circ})\) 49. \(\cos (18x)\)
E: Verify identities
Exercise \(\PageIndex{E}\)
\( \bigstar \) Prove the identity.
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51. \((\sin^2x-1)^2=\cos(2x)+\sin^4x \\[4pt]\) 52. \((\sin t-\cos t)^2=1-\sin(2t) \\[4pt]\) 53. \(\sin(2x)=-2 \sin(-x) \cos(-x) \\[4pt]\) 54. \(\cot x-\tan x=2 \cot(2x) \\[4pt]\) 55. \(\dfrac{1+\cos (2\theta )}{\sin (2\theta )}\tan ^2\theta =\tan \theta \\[4pt]\) 56. \(\dfrac{1+\cos (2t)}{\sin (2t)-\cos t}=\dfrac{2\cos t}{2\sin t-1} \) |
57. \(\sin (2x)=\dfrac{2\tan x}{1+\tan ^2x} \\[4pt]\) 58. \(\cos (2\alpha )=\dfrac{1-\tan ^2\alpha }{1+\tan ^2\alpha } \\[4pt] \\[4pt]\) 59. \(\tan (2x)=\dfrac{2\sin x \cos x }{2\cos ^2 x-1}\) 60. \(\dfrac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)} =\tan \left(\theta \right)\) 61. \(\cos(16x) = \Big( \cos^2(4x)-\sin^2(4x)-\sin(8x) \Big) \\ \) 62. \(\sin(16x)=16 \sin x \cos x \cos(2x)\cos(4x)\cos(8x) \\[4pt]\) |
- Answers to odd exercises.
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51. \((\sin^2x-1)^2 = \sin^4 x - 2 \sin^2 x + 1 = \cos(2x)+\sin^4x \)
53. \(-2 \sin(-x)\cos(-x)=-2(-\sin(x)\cos(x))=\sin(2x)\)
55. \(\dfrac{\sin (2\theta )}{1+\cos (2\theta )}\tan ^2\theta =\dfrac{2\sin (\theta )\cos (\theta )}{1+\cos ^2\theta -\sin ^2\theta }\tan ^2\theta=\dfrac{2\sin (\theta )\cos (\theta )}{2\cos ^2\theta }\tan ^2\theta=\dfrac{\sin (\theta )}{\cos (\theta )}\tan ^2\theta \\
=\cot (\theta )\tan ^2\theta=\tan \theta\)57. \(\dfrac{2\tan x}{1+\tan ^2x}=\dfrac{\tfrac{2\sin x}{\cos x}}{1+\tfrac{\sin ^2x}{\cos ^2x}}=\dfrac{\tfrac{2\sin x}{\cos x}}{\tfrac{\cos ^2x+\sin ^2x}{\cos ^2x}}=\dfrac{2\sin x}{\cos x}\cdot \dfrac{\cos ^2x}{1}=2\sin x \cos x=\sin (2x)\)
59. \(\dfrac{2\sin x \cos x }{2\cos ^2 x-1}=\dfrac{\sin (2x)}{ \cos (2x)}=\tan (2x)\)
61. \( (\cos^2(4x)-\sin^2(4x)-\sin(8x))(\cos^2(4x)-\sin^2(4x)+\sin(8x)) \\
= (\cos(8x)-\sin(8x))(\cos(8x)+\sin(8x)) = \cos ^2 (8x)-\sin ^2 (8x)= \cos(16x) \)
F: Use Power Reduction Formulas
Exercise \(\PageIndex{F}\)
\( \bigstar \) Rewrite the expression with an exponent no higher than 1.
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70. \(\cos ^2 (5x) \\[4pt]\) 71. \(\cos ^2 (6x) \\[4pt]\) 72. \(\sin ^4 (8x) \\[4pt]\) 73. \(\sin ^4 (3x)\) |
74. \(\sin^2(2x) \\[4pt]\) 75. \(\sin^2x \cos^2x \\[4pt]\) 76. \(\tan^2x \sin^2x \\[4pt]\) 77. \(\tan^4x\) |
78. \(\cos^2x \sin (2x) \\[4pt]\) 79. \(\tan^4x \cos^2 x \\[4pt] \) 80. \(\tan ^2\left ( \dfrac{x}{2} \right )\sin x \\[4pt]\) 81. \(\cos^2(2x) \sin x \) |
82. \(\tan^2x \sin x \\[4pt]\) 83. \(\cos^4x \sin^2x \\[4pt]\) 84. \(\cos^2x \sin^4x \\[4pt]\) |
\( \bigstar \) Algebraically find an equivalent function, in terms of only \(\sin x\) and/or \(\cos x\)
| 85. \(\sin (4x)\) | 86. \(\cos (4x)\) |
- Answers to odd exercises.
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71. \(\dfrac{1+\cos (12x)}{2}\) 73. \(\dfrac{3+\cos(12x)-4\cos(6x)}{8}\) 75. \(\dfrac{1-\cos(4x)}{8}\)
77. \(\dfrac{3+\cos(4x)-4\cos(2x)}{3+\cos(4x)+4\cos(2x)}\) 79. \(\dfrac{3+\cos(4x)-4\cos(2x)}{4(\cos(2x)+1)}\) 81. \(\dfrac{(1+\cos(4x))\sin x}{2}\)
83. \(\dfrac{2+\cos(2x)-2\cos(4x)-\cos(6x)}{32}\) 85. \(4\sin x\cos x(\cos^2x-\sin^2x)\)
G: Use Half Angle Formulas
Exercise \(\PageIndex{G}\)
\( \bigstar \) Find the exact value using half-angle formulas.
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91. \(\sin \left ( \dfrac{\pi }{8} \right ) \\[4pt]\) 92. \(\cos \left ( -\dfrac{11\pi }{12} \right )\) |
93. \(\sin \left ( \dfrac{11\pi }{12} \right ) \\[4pt]\) 94. \(\cos \left ( \dfrac{7\pi }{8} \right )\) |
95. \(\tan \left ( \dfrac{5\pi }{12} \right ) \\[4pt]\) 96. \(\tan \left ( -\dfrac{3\pi }{12} \right )\) |
97. \(\tan \left ( -\dfrac{3\pi }{8} \right ) \\[4pt]\)
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\( \bigstar \) Find the exact values of a) \(\sin \left ( \dfrac{x}{2} \right )\) b) \(\cos \left ( \dfrac{x}{2} \right )\) and c) \(\tan \left ( \dfrac{x}{2} \right )\) when \(0 \leq \theta \leq 2\pi \)
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101. If \(\sin x =-\dfrac{12}{13}\), and \(x\) is in quadrant \(\mathrm{III} \\[4pt] \). 102. If \(\tan x =-\dfrac{4}{3}\), and \(x\) is in quadrant \(\mathrm{IV} \). |
103. If \(\sec x =-4\), and \(x\) is in quadrant \(\mathrm{II} \\[4pt] \). 104. If \(\csc x =7\), and \(x\) is in quadrant \(\mathrm{II} \). |
\( \bigstar \) Use the figure below to find the requested half angle trigonometric expressions.
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107. Find \(\sin \left (\dfrac{\alpha }{2} \right )\), \(\cos \left (\dfrac{\alpha }{2} \right )\), 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.
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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. \(\dfrac{2\sqrt{13}}{13}, \dfrac{3\sqrt{13}}{13}, \dfrac{2}{3}\)
\( \bigstar \)