Simplifying $\sin^{-1}{\left(\frac{\sqrt{3}}{2}\right)}$

Question

I was wondering how you would simplify this inverse trig function.

$$\sin^{-1}{\left(\frac{\sqrt{3}}{2}\right)}$$

Is there a way to do it without using a calculator? or is calculator the best approach.

Thanks

Answer 1

There is nothing to simplify. You just need to know the fact that $$\sin{\frac{\pi}{3}}=\frac{\sqrt{3}}{2}$$

Therefore $$\sin^{-1}{\left(\frac{\sqrt{3}}{2}\right)}=\frac{\pi}{3}$$

The answer is unique because, as you might remember, there is only one value on the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ (the range of the inverse sine function) where $\sin{x}$ equals $\frac{\sqrt{3}}{2}$ and that, of course, is $\frac{\pi}{3}$.