Solve the equations:

Give a general formula for all the solutions and list all the solutions in the interval $[ -2\pi ,2\pi ] .$

62

Figure 89

Solution (a) Use the inverse sine function $y =\sin ^{-1} x$, $-\dfrac{\pi}{2}\leq y\leq \dfrac{\pi}{2}$. $$\begin{array}{l} \sin \theta = \frac{1}{2} \\ \,\,\,\theta = \sin ^{ - 1} \frac{1}{2}\, \quad {\color{#0066A7}{- \frac{\pi }{2} \le \theta \le \frac{\pi }{2}}} \\ \,\,\theta = \frac{\pi }{6} \\ \end{array}$$

Over the interval $[ 0,2\pi ] ,$ there are two angles $\theta$ for which $\sin \theta =\dfrac{1}{2}$. See Figure 89.

All the solutions of $\sin \theta =\dfrac{1}{2}$ are given by the general formula $$\theta =\dfrac{\pi }{6}+2k\pi\quad {\rm \ or \ }\quad \theta =\dfrac{5\pi }{6}+2k\pi,\qquad {\rm where }\ k \ \hbox{is any integer}$$

The solutions in the interval $[-2\pi, 2\pi]$ are $$\left\{ -\dfrac{11\pi }{6},-\dfrac{7\pi }{6},\dfrac{\pi }{6 },\dfrac{5\pi }{6}\right\}$$

(b) A calculator must be used to solve $\cos \theta =0.4$. Then $$\theta =\cos ^{-1}\left( 0.4\right) \approx 1.159279 \quad 0\leq \theta \leq \pi$$

Rounded to three decimal places, $\theta \,{=}\,\cos ^{-1}0.4=1.159$ radians. But there is another angle $\theta$ in the interval $[ 0,2\pi ]$ for which $\cos \theta =0.4$, namely, $\theta \approx 2\pi -1.159$ $\approx 5.124$ radians.

Because the cosine function has period $2\pi$, all the solutions of $\cos \theta =0.4$ are given by the general formulas $$\theta \approx 1.159+2k\pi \quad \hbox{or} \quad \theta \approx 5.124+2k\pi,\qquad \hbox{ where } k \hbox{ is any integer}$$

The solutions in the interval $[ -2\pi ,2\pi ]$ are $\left\{ -5.124,-1.159, 1.159, 5.124\right\}$.