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27.4: Review of trigonometric functions

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    68485

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    Exercise \(\PageIndex{1}\)

    Fill in all the trigonometric function values in the table below.

    \[\begin{array}{c||c|c|c|c|c}
    & 0 & \dfrac{\pi}{6} & \dfrac{\pi}{4} & \dfrac{\pi}{3} & \dfrac{\pi}{2} \\
    \hline \hline \sin (x) & & & & & \\
    \hline \cos (x) & & & & & \\
    \hline \tan (x) & & & & &
    \end{array} \nonumber \]

    Answer

    \(\begin{array}{c||c|c|c|c|c}
    x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
    \hline \hline \sin (x) & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{3}}{2} & 1 \\
    \hline \cos (x) & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0 \\
    \hline \tan (x) & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text { undef. }
    \end{array}\)

    Exercise \(\PageIndex{2}\)

    Find the exact values of

    1. \(\cos\left(-\dfrac{\pi}{6}\right)\)
    2. \(\sin\left(-\dfrac{\pi}{4}\right)\)
    3. \(\tan\left(-\dfrac{\pi}{3}\right)\)
    Answer
    1. \(\dfrac{\sqrt{3}}{2}\)
    2. \(\dfrac{-\sqrt{2}}{2}\)
    3. \(-\sqrt{3}\)

    Exercise \(\PageIndex{3}\)

    Find the exact value of the trigonometric function.

    1. \(\sin\left(\dfrac{5\pi}{4}\right)\)
    2. \(\cos\left(\dfrac{11\pi}{6}\right)\)

    [Hint: Use the special \(45^\circ-45^\circ-90^\circ\) or \(30^\circ-60^\circ-90^\circ\) triangles to find the solution.]

    Answer
    1. \(\dfrac{-\sqrt{2}}{2}\)
    2. \(\dfrac{\sqrt{3}}{2}\)

    Exercise \(\PageIndex{4}\)

    Find the amplitude, period, and the phase shift of the given function. Draw the graph over a one-period interval. Label all maxima, minima and intercepts.

    1. \(y=3 \cos\left(4 x-\pi\right)\)
    2. \(y=-5\sin\left(x+\dfrac{\pi}{2}\right)\)
    Answer
    1. amplitude \(3\), period \(\dfrac{\pi}{2}\), phase-shift \(\dfrac{\pi}{4}\) clipboard_eae7381b7075b464c7074552724f5f1ed.png
    2. amplitude \(5\), period \(2\pi \), phase-shift \(\dfrac{-\pi}{2}\) clipboard_e24e4d7c331858767be9f4c23ff7a2c75.png

    Exercise \(\PageIndex{5}\)

    Find the exact trigonometric function value.

    1. \(\cos\left(\dfrac{\pi}{12}\right)\) [Hint: Use the addition and subtraction of angles formulas.]
    2. \(\cos\left(\dfrac{3\pi}{8}\right)\) [Hint: Use the half-angles formulas.]
    Answer
    1. \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
    2. \(\dfrac{\sqrt{2-\sqrt{2}}}{2}\)

    Exercise \(\PageIndex{6}\)

    Let \(\sin(\alpha)=-\dfrac{4}{5}\) and let \(\alpha\) be in quadrant III. Find \(\sin(2\alpha)\), \(\cos(2\alpha)\), and \(\tan(2\alpha)\).

    Answer

    \(\sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)

    Exercise \(\PageIndex{7}\)

    Find the exact value of:

    1. \(\sin^{-1}\left(-\dfrac{1}{2}\right)\)
    2. \(\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\)
    3. \(\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)\)
    Answer
    1. \(\dfrac{-\pi}{6}\)
    2. \(\dfrac{5 \pi}{6}\)
    3. \(\dfrac{-\pi}{6}\)

    Exercise \(\PageIndex{8}\)

    Solve for \(x\): \(2\sin(x)+\sqrt{3}=0\)

    Answer

    \(x=(-1)^{n+1} \dfrac{\pi}{3}+n \pi\), where \(n=0, \pm 1, \ldots \)

    Exercise \(\PageIndex{9}\)

    Solve for \(x\): \(\tan^2(x)-1=0\)

    Answer

    \(x=\pm \dfrac{\pi}{4}+n \pi \) where \(n=0, \pm 1, \ldots \)

    Exercise \(\PageIndex{10}\)

    Solve for \(x\).

    1. \(2\cos^2(x)-1=0\)
    2. \(2\sin^2(x)+15\sin(x)+7=0\)
    Answer
    1. \(x=\pm \dfrac{\pi}{4}+2 n \pi\) or \(x=\pm \dfrac{3 \pi}{4}+2 n \pi\) where \(n=0, \pm 1, \ldots\)
    2. \((-1)^{n+1} \dfrac{\pi}{6}+n \pi\) where \(n=0, \pm 1, \ldots\)

    This page titled 27.4: Review of trigonometric functions 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.