Inverse sine function

The inverse sine of $ u$, denoted by  $ \sin^{-1} u$, is defined as the unique value $ -\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}$ satisfying $ \sin\theta = u$. The inverse sine function `` $ \sin^{-1} x$'' is also known as the arcsine function ``$ \arcsin x$.''

\includegraphics{lec14a.ps}

Domain and range of $ \sin^{-1}$. $ D = [-1,\:1]$ and $ R = \left[-\dfrac{\pi}{2},\:\dfrac{\pi}{2}\right]$.

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$ $ ^\dagger$ as

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

$ \dagger$ When $ u = 1$ or $ u = -1$, we find $ \theta_1 = \theta_2$; thus, there is only one solution in the interval $ [0,2\pi)$.



Department of Mathematics
Last modified: 2005-09-29