Skip to main content
Mathematics LibreTexts

7.3E: Double Angle Identities (Exercises)

  1. Last updated
  2. Save as PDF
  • Page ID
    13936

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Section 7.3 Exercises

    1. If \(\sin \left(x\right)=\dfrac{1}{8}\) and \(x\) is in quadrant I, then find exact values for (without solving for \(x\)):

    a. \(\sin \left(2x\right)\)
    b. \(\cos \left(2x\right)\)
    c. \(\tan \left(2x\right)\)

    2. If \(\cos \left(x\right)=\dfrac{2}{3}\) and \(x\) is in quadrant I, then find exact values for (without solving for \(x\)):

    a. \(\sin \left(2x\right)\)
    b. \(\cos \left(2x\right)\)
    c. \(\tan \left(2x\right)\)

    Simplify each expression.

    3. \(\cos ^{2} \left(28{}^\circ \right)-\sin ^{2} (28{}^\circ )\)

    4. \(2\cos ^{2} \left(37{}^\circ \right)-1\)

    5. \(1-2\sin ^{2} (17{}^\circ )\)

    6. \(\cos ^{2} \left(37{}^\circ \right)-\sin ^{2} (37{}^\circ )\)

    7. \(\cos ^{2} \left(9x\right)-\sin ^{2} (9x)\)

    8. \(\cos ^{2} \left(6x\right)-\sin ^{2} (6x)\)

    9. \(4\sin \left(8x\right){\rm cos}(8x)\)

    10. \(6\sin \left(5x\right){\rm cos}(5x)\)

    Solve for all solutions on the interval \([0, 2\pi )\).

    11. \(6\sin \left(2t\right)+9\sin \left(t\right)=0\)

    12. \(2\sin \left(2t\right)+3\cos \left(t\right)=0\)

    13. \(9\cos \left(2\theta \right)=9\cos ^{2} \left(\theta \right)-4\)

    14. \(8\cos \left(2\alpha \right)=8\cos ^{2} \left(\alpha \right)-1\)

    15. \(\sin \left(2t\right)=\cos \left(t\right)\)

    16. \(\cos \left(2t\right)=\sin \left(t\right)\)

    17. \(\cos \left(6x\right)-\cos \left(3x\right)=0\)

    18. \(\sin \left(4x\right)-\sin \left(2x\right)=0\)

    Use a double angle, half angle, or power reduction formula to rewrite without exponents.

    19. \(\cos ^{2} (5x)\)

    20. \(\cos ^{2} (6x)\)

    21. \(\sin ^{4} (8x)\)

    22. \(\sin ^{4} \left(3x\right)\)

    23. \(\cos ^{2} x\sin ^{4} x\)

    24. \(\cos ^{4} x\sin ^{2} x\)

    25. If \(\csc \left(x\right)=7\) and \(90{}^\circ <x<180{}^\circ\), then find exact values for (without solving for \(x\)):

    a. \(\sin \left(\dfrac{x}{2} \right)\)
    b. \(\cos \left(\dfrac{x}{2} \right)\)
    c. \(\tan \left(\dfrac{x}{2} \right)\)

    26. If \(\sec \left(x\right)=4\) and \(270{}^\circ <x<360{}^\circ\), then find exact values for (without solving for \(x\)):

    a. \(\sin \left(\dfrac{x}{2} \right)\)
    b. \(\cos \left(\dfrac{x}{2} \right)\)
    c. \(\tan \left(\dfrac{x}{2} \right)\)

    Prove the identity.

    27. \(\left(\sin t-\cos t\right)^{2} =1-\sin \left(2t\right)\)

    28. \(\left(\sin ^{2} x-1\right)^{2} =\cos \left(2x\right)+\sin ^{4} x\)

    29. \(\sin \left(2x\right)=\dfrac{2\tan \left(x\right)}{1+\tan ^{2} \left(x\right)}\)

    30. \(\tan \left(2x\right)=\dfrac{2\sin \left(x\right)\cos \left(x\right)}{2\cos ^{2} \left(x\right)-1}\)

    31. \(\cot \left(x\right)-\tan \left(x\right)=2\cot \left(2x\right)\)

    32. \(\dfrac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)} =\tan \left(\theta \right)\)

    33. \(\cos \left(2\alpha \right)=\dfrac{1-\tan ^{2} \left(\alpha \right)}{1+\tan ^{2} \left(\alpha \right)}\)

    34. \(\dfrac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos \left(t\right)} =\dfrac{2\cos \left(t\right)}{2\sin \left(t\right)-1}\)

    35. \(\sin \left(3x\right)=3\sin \left(x\right)\cos ^{2} \left(x\right)-\sin ^{3} (x)\)

    36. \(\cos \left(3x\right)=\cos ^{3} (x)-3\sin ^{2} (x)\cos \left(x\right)\)

    Answer

    1. a. \(\dfrac{3\sqrt{7}}{32}\)
    b. \(\dfrac{31}{32}\)
    c. \(\dfrac{3\sqrt{7}}{31}\)

    3. \(\cos(56^{\circ})\)

    5. \(\cos(34^{\circ})\)

    7. \(\cos(18x)\)

    9. \(2\sin(16x)\)

    11. 0, \(\pi\), 2.4189,3.8643

    13. 0.7297, 2.4119, 3.8713, 5.5535

    15. \(\dfrac{\pi}{6}\), \(\dfrac{\pi}{2}\), \(\dfrac{5\pi}{6}\), \(\dfrac{3\pi}{2}\)

    17. a. \(\dfrac{2\pi}{9}\), \(\dfrac{4\pi}{9}\), \(\dfrac{8\pi}{9}\), \(\dfrac{10\pi}{9}\), \(\dfrac{14\pi}{9}\), \(\dfrac{16\pi}{9}\), 0, \(\dfrac{2\pi}{3}\), \(\dfrac{4\pi}{3}\)

    19. \(\dfrac{1 + \cos(10x)}{2}\)

    21. \(\dfrac{3}{8} - \dfrac{1}{2} \cos(16x) + \dfrac{1}{8} \cos(32x)\)

    23. \(\dfrac{1}{16} - \dfrac{1}{16} \cos(2x) + \dfrac{1}{16} \cos(4x) - \dfrac{1}{16} \cos(2x) \cos(4x)\)

    25. a. \(\sqrt{\dfrac{1}{2}+\dfrac{2 + \sqrt{7}}{7}}\)
    b. \(\sqrt{\dfrac{1}{2}-\dfrac{2 + \sqrt{7}}{7}}\)
    c. \(\dfrac{1}{7 - 4\sqrt{3}}\)

    This page titled 7.3E: Double Angle Identities (Exercises) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by David Lippman & Melonie Rasmussen (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.