Question

Find the exact value of ${\sin }^{-1}(-0.5)$.

Answer 1

Find the value of ${\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\left(-0.5\right)$:

It has known that the value of ${\sin }^{-1}\left(0.5\right)$ is $\frac{\pi }{6}.$

The value of ${\sin }^{-1}\left(x\right)$ is the range between $\frac{-\pi }{2}$and $\frac{\pi }{2}$.

Calculate the ${\sin }^{-1}\left(-0.5\right)$ value.

Consider that, $\begin{array}{rcl}t& =& {\sin }^{-1}(-0.5)\end{array}$

Shift ${\sin }^{-1}$ to the left side, so it becomes $\sin$:

$\begin{array}{rcl}\sin t& =& (-0.5)\end{array}$

Substitute the value of ${\sin }^{-1}\left(0.5\right)=\frac{\pi }{6}$

$\begin{array}{rcl}-0.5& =& -\sin \left(\frac{\pi }{6}\right)\\ & =& \sin \left(\frac{-\pi }{6}\right)\end{array}$

Hence, the value of ${\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\left(-0.5\right)$ will be $\frac{\mathbf{-}\mathbf{\pi }}{\mathbf{6}}$.