Skip to main content
Mathematics LibreTexts

18.3: Exercises

  • Page ID
    49068
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Exercise \(\PageIndex{1}\)

    Find the trigonometric function values.

    1. \(\sin\left(\dfrac{5\pi}{12}\right)\)
    2. \(\cos\left(\dfrac{5\pi}{12}\right)\)
    3. \(\tan\left(\dfrac{\pi}{12}\right)\)
    4. \(\sin\left(\dfrac{7\pi}{12}\right)\)
    5. \(\cos\left(\dfrac{11\pi}{12}\right)\)
    6. \(\sin\left(\dfrac{2\pi}{3}\right)\)
    7. \(\sin\left(\dfrac{5\pi}{6}\right)\)
    8. \(\cos\left(\dfrac{3\pi}{4}\right)\)
    9. \(\tan\left(\dfrac{13\pi}{12}\right)\)
    10. \(\cos\left(-\dfrac{\pi}{12}\right)\)
    11. \(\sin\left(\dfrac{11\pi}{12}\right)\)
    12. \(\sin\left(\dfrac{29\pi}{12}\right)\)
    Answer
    1. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
    2. \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
    3. \(2-\sqrt{3}\)
    4. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
    5. \(\dfrac{-(\sqrt{2}+\sqrt{6})}{4}\)
    6. \(\dfrac{\sqrt{3}}{2}\)
    7. \(\dfrac{1}{2}\)
    8. \(\dfrac{-\sqrt{2}}{2}\)
    9. \(2-\sqrt{3}\)
    10. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
    11. \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
    12. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)

    Exercise \(\PageIndex{2}\)

    Simplify the function \(f\) using the addition and subtraction formulas.

    1. \(f(x)=\sin\left(x+\dfrac{\pi}{2}\right)\)
    2. \(f(x)=\cos\left(x-\dfrac{\pi}{4}\right)\)
    3. \(f(x)=\tan\left(\pi-x\right)\)
    4. \(f(x)=\sin\left(\dfrac{\pi}{6}-x\right)\)
    5. \(f(x)=\cos\left(x+\dfrac{11\pi}{12}\right)\)
    6. \(f(x)=\cos\left(\dfrac{2\pi}{3}-x\right)\)
    Answer
    1. \(\sin \left(x+\dfrac{\pi}{2}\right)=\cos (x)\)
    2. \(\cos \left(x-\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}(\sin (x)+\cos (x))\)
    3. \(\tan (\pi-x)=-\tan (x)\)
    4. \(\sin \left(\dfrac{\pi}{6}-x\right)=\dfrac{1}{2} \cos (x)-\dfrac{\sqrt{3}}{2} \sin (x)\)
    5. \(\cos \left(x+\dfrac{11 \pi}{12}\right)=\left(\dfrac{-(\sqrt{2}+\sqrt{6})}{4}\right) \cdot \cos (x)-\left(\dfrac{\sqrt{6}-\sqrt{2}}{4}\right) \cdot \sin (x)\)
    6. \(\cos \left(\dfrac{2 \pi}{3}-x\right)=\dfrac{-1}{2} \cdot \cos (x)+\dfrac{\sqrt{3}}{2} \cdot \sin (x)\)

    Exercise \(\PageIndex{3}\)

    Find the exact values of the trigonometric functions applied to the given angles by using the half-angle formulas.

    1. \(\cos\left(\dfrac{\pi}{12}\right)\)
    2. \(\tan\left(\dfrac{\pi}{8}\right)\)
    3. \(\sin\left(\dfrac{\pi}{24}\right)\)
    4. \(\cos\left(\dfrac{\pi}{24}\right)\)
    5. \(\sin\left(\dfrac{9\pi}{8}\right)\)
    6. \(\tan\left(\dfrac{3\pi}{8}\right)\)
    7. \(\sin\left(\dfrac{-\pi}{8}\right)\)
    8. \(\sin\left(\dfrac{13\pi}{24}\right)\)
    Answer
    1. \(\dfrac{\sqrt{2+\sqrt{3}}}{2}=\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
    2. \(\sqrt{2}-1\)
    3. \(\dfrac{\sqrt{8-2 \sqrt{2}-2 \sqrt{6}}}{4}\)
    4. \(\dfrac{\sqrt{8+2 \sqrt{2}+2 \sqrt{6}}}{4}\)
    5. \(-\dfrac{\sqrt{2-\sqrt{2}}}{2}\)
    6. \(1+\sqrt{2}\)
    7. \(-\dfrac{\sqrt{2-\sqrt{2}}}{2}\)
    8. \(\dfrac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}\)

    Exercise \(\PageIndex{4}\)

    Find the exact values of the trigonometric functions of \(\dfrac \alpha 2\) and of \(2\alpha\) by using the half-angle and double angle formulas.

    1. \(\sin(\alpha)=\dfrac{4}{5}\), and \(\alpha\) in quadrant I
    2. \(\cos(\alpha)=\dfrac{7}{13}\), and \(\alpha\) in quadrant IV
    3. \(\sin(\alpha)=\dfrac{-3}{5}\), and \(\alpha\) in quadrant III
    4. \(\tan(\alpha)=\dfrac{4}{3}\), and \(\alpha\) in quadrant III
    5. \(\tan(\alpha)=\dfrac{-5}{12}\), and \(\alpha\) in quadrant II
    6. \(\cos(\alpha)=\dfrac{-2}{3}\), and \(\alpha\) in quadrant II
    Answer
    1. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{5}}{5}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{2 \sqrt{5}}{5}, \tan \left(\dfrac{\alpha}{2}\right)=\dfrac{1}{2}, \sin (2 \alpha)={\dfrac{24}{25}}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)
    2. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{39}}{13}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{130}}{13}, \tan \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{30}}{10}, \sin (2 \alpha)=\dfrac{-28 \sqrt{30}}{169}, \cos (2 \alpha)=\dfrac{-71}{169}, \tan (2 \alpha)=\dfrac{28 \sqrt{30}}{71} \)
    3. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{3 \sqrt{10}}{10}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{10}}{10}, \tan \left(\dfrac{\alpha}{2}\right)=-3, \sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{7}{25}, \tan (2 \alpha)=\dfrac{24}{7}\)
    4. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{2 \sqrt{5}}{5}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{5}}{5}, \tan \left(\dfrac{\alpha}{2}\right)=-2, \sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)
    5. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{5 \sqrt{26}}{26}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{26}}{26}, \tan \left(\dfrac{\alpha}{2}\right)=5, \sin (2 \alpha)=\dfrac{-120}{169}, \cos (2 \alpha)=\dfrac{119}{169}, \tan (2 \alpha)=\dfrac{-120}{119}\)
    6. \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{30}}{6}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{6}}{6}, \tan \left(\dfrac{\alpha}{2}\right)=\sqrt{5}, \sin (2 \alpha)=\dfrac{-4 \sqrt{5}}{9}, \cos (2 \alpha)=\dfrac{-1}{9}, \tan (2 \alpha)=4 \sqrt{5}\)

    This page titled 18.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.