Question

In Problems 79-90, find the exact value of each expression.

$\sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]$

Short answer

The exact value is$\frac{\sqrt{3}}{2}$

Step-by-step solution

Given information

Given expression$\sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]$

Rewriting using implicit form

Rewriting, we get

$$\begin{gathered} \sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]=\sin \left[2{\sin }^{-1}\right(\frac{\mathrm{\pi }}{3}\left)\right] \\ \sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]=\sin [2\times {60}^{\circ }] \\ \sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]=\sin {120}^{\circ } \\ \sin \left[2{\sin }^{-1}\frac{\sqrt{3}}{2}\right]=\frac{\sqrt{3}}{2} \end{gathered}$$