27.4: Review of trigonometric functions

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 $\dfrac{\pi}{4}$
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.]

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 $n=0, \pm 1, \ldots$