
# 7.3E: Exercises: Double Angle, Half Angle and Reductions Formulas


## 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 $$0$$.)

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 $$\PageIndex{B}$$

 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 $$\pi \leq \theta \leq \frac{3\pi}{2}$$

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 on the interval $$[0,2\pi )$$.

 20. $$\sin(2t)=\cos t \\[2pt]$$ 21. $$\sin(2t)+ \sin t=0 \\[2pt]$$ 22. $$\sin(2x)-\sin x=0 \\[2pt]$$ 23. $$\sin (t) +3\cos (t) =0 \\[2pt]$$ 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. $$3\cos (2\alpha )=2\cos ^{2} (\alpha ) \\[2pt]$$ 31. $$\cos(2x)-\cos x=0 \\[2pt]$$ 32. $$4\sin^2 x+\sin(2x)\sec x-3=0 \\[2pt]$$ 33. $$\sin^2 x-1+2\cos(2x)-\cos^2 x=1$$

21. $$0, \dfrac{2\pi }{3}, \pi , \dfrac{4\pi }{3}$$     23. $$\dfrac{\pi }{2}, \dfrac{3\pi }{2}$$    25. $$0, \dfrac{\pi }{2}, \pi, \dfrac{3\pi }{2}$$
27. $$\dfrac{3\pi }{2}, \dfrac{\pi }{6}, \dfrac{5\pi }{6}$$     29. $$\dfrac{\pi }{6}, \dfrac{5\pi }{6}, \dfrac{7\pi }{6}, \dfrac{11\pi }{6}$$     31. $$0, \dfrac{2\pi }{3}, \dfrac{4\pi }{3}$$     33. No solution.

### D: Recognize patterns

Exercise $$\PageIndex{D}$$

$$\bigstar$$  Simplify to one trigonometric expression. Do not evaluate.

 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. $$\cos ^{2} \left(37{}^\circ \right)-\sin ^{2} (37{}^\circ ) \\[4pt]$$

41.  $$2\sin \left ( \dfrac{\pi }{2} \right )$$     43. $$3\sin (10x)$$     45. $$\cos (134^{\circ})$$       47. $$\cos (74^{\circ})$$   49. $$\cos (18x)$$

### E: Verify identities

Exercise $$\PageIndex{E}$$

$$\bigstar$$  Prove the identity.

 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. $$\cos(16x) = \Big( \cos^2(4x)-\sin^2(4x)-\sin(8x) \Big) \\$$                            $$\cdot \; \Big( \cos^2(4x)-\sin^2(4x) + \sin(8x) \Big)$$ 62. $$\sin(16x)=16 \sin x \cos x \cos(2x)\cos(4x)\cos(8x) \\[4pt]$$

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. $$(\cos^2(4x)-\sin^2(4x)-\sin(8x))(\cos^2(4x)-\sin^2(4x)+\sin(8x)) \\ = (\cos(8x)-\sin(8x))(\cos(8x)+\sin(8x)) = \cos ^2 (8x)-\sin ^2 (8x)= \cos(16x)$$

### D: Use Power Reduction Formulas

Exercise $$\PageIndex{D}$$

$$\bigstar$$  Rewrite the expression with an exponent no higher than 1.

 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. $$\cos (4x)$$

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. $$4\sin x\cos x(\cos^2x-\sin^2x)$$

### A: Use Half Angle Formulas

Exercise $$\PageIndex{A}$$

$$\bigstar$$  Find the exact value using half-angle formulas.

 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. $$\tan \left ( -\dfrac{3\pi }{8} \right ) \\[4pt]$$

$$\bigstar$$ 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 $$0 \leq \theta \leq 2\pi$$

 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.

 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 $$\tan \left (\dfrac{\theta }{2} \right )\\[4pt]$$.
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. $$\dfrac{2\sqrt{13}}{13}, \dfrac{3\sqrt{13}}{13}, \dfrac{2}{3}$$
$$\bigstar$$