# 27.4: Review of trigonometric functions

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## Exercise $$\PageIndex{1}$$

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

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

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

## Exercise $$\PageIndex{2}$$

Find the exact values of

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

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

## 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.]
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 $$\tan(2\alpha)$$.

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

## 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. $$\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)$$
1. $$\dfrac{-\pi}{6}$$
2. $$\dfrac{5 \pi}{6}$$
3. $$\dfrac{-\pi}{6}$$

## Exercise $$\PageIndex{8}$$

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

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

## Exercise $$\PageIndex{9}$$

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

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

## Exercise $$\PageIndex{10}$$

Solve for $$x$$.

1. $$2\cos^2(x)-1=0$$
2. $$2\sin^2(x)+15\sin(x)+7=0$$
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 $$n=0, \pm 1, \ldots$$