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7.3e: Exercises: Double Angle, Half Angle and Reductions Formulas

  • Page ID
    74636
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    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.

    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 \( \pi \leq \theta \leq \frac{3\pi}{2} \)
    Answers to odd exercises.

    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 \([0,2\pi )\).

    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.

    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.

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

     

    Answers to odd exercises.

    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.

    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. \(\sin(16x)=16 \sin x \cos x \cos(2x)\cos(4x)\cos(8x)   \\[4pt]\)

    Answers to odd exercises.

    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.

    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.

    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.

    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.

    CNX_Precalc_Figure_07_03_201.jpg

    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.

    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. \(\dfrac{2\sqrt{13}}{13}, \dfrac{3\sqrt{13}}{13}, \dfrac{2}{3}\)

    \( \bigstar \)


    7.3e: 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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