Precalculus Review

Inverse sine function

The inverse sine of

, denoted by $\sin^{-1} u$ , is defined as the unique value $-\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}$ satisfying $\sin\theta = u$ . It is also known as the arcsine function $\arcsin x$ . The domain and the range are given by $D = [-1,\:1]$ and $R = \left[-\dfrac{\pi}{2},\:\dfrac{\pi}{2}\right]$ .

$\includegraphics{lec14a.ps}$

Trigonometric equation. Consider the solutions in the cycle $[0,2\pi)$ to the equation ${\sin\theta = u}$ . By observing $\sin(\pi - \theta) = -\sin(-\theta) = \sin\theta$ , we obtain the two solutions $\theta_1$ and $\theta_2$ as

$\begin{displaymath} \begin{array}{lll} \theta_1 = \sin^{-1} u & \mbox{and}\qua... ... 2\pi + \sin^{-1} u & \quad\mbox{ if $u < 0$. } \end{array} \end{displaymath}$

When

, we find that $\theta_1 = \theta_2$ , and therefore, that there is only one solution in the interval $[0,2\pi)$ . Also note that if

then $\sin^{-1} u < 0$ is out of the interval $[0,2\pi)$ . Once the two solutions $\theta_1$ and $\theta_2$ in the interval $[0,2\pi)$ to the trigonometric equation are obtained, the general solutions are given by

$\displaystyle \theta = \theta_1 + 2\pi n,\: \theta_2 + 2\pi n$ for every integer

. $\displaystyle$