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Mathematics LibreTexts

27.4: Review of trigonometric functions

  1. CCBYNCSA
  2. Last updated
    May 2, 2022
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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.

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