Question

Find the exact solution of the equation.

$-3{\sin }^{-1}x=\mathrm{\pi }$

Short answer

The exact solution of the equation is $\left\{-\frac{\sqrt{3}}{2}\right\}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$-3{\sin }^{-1}x=\mathrm{\pi }$

Divide both sides by $-3$,

$$\begin{gathered} \frac{-3{\sin }^{-1}x}{-3}=\frac{\mathrm{\pi }}{-3} \\ {\sin }^{-1}x=-\frac{\mathrm{\pi }}{3} \end{gathered}$$

As $y={\sin }^{-1}x$means $x=\sin y$, where,

$-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

Step 2. Rewrite the equation.

On rewriting the equation, we get,

$$\begin{gathered} x=\sin \left(-\frac{\mathrm{\pi }}{3}\right) \\ x=-\sin \left(\frac{\mathrm{\pi }}{3}\right) \\ x=-\frac{\sqrt{3}}{2} \end{gathered}$$

Therefore, the solution set isrole="math" localid="1646497501412" $\left\{-\frac{\sqrt{3}}{2}\right\}$.