Question

In Problems 13–24, find the exact value of each expression.

${\sin }^{-1}\frac{\sqrt{2}}{2}$

Short answer

The exact value of ${\sin }^{-1}\frac{\sqrt{2}}{2}is\frac{\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given Information:   

Given expression:${\sin }^{-1}\frac{\sqrt{2}}{2}$

We want to find the exact value of given expression.

Step 2. Solution: 

The given expression can be rewritten as:

${\sin }^{-1}\frac{\sqrt{2}}{2}=y;-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$

Hence we get $\sin y=\frac{\sqrt{2}}{2}$

Since we have sin^-1 function initially, the bounds on the sin function are going to be in the interval $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$ which represents the range of sin^-1 function.

Using the unit circle, we know that in order to get $\frac{\sqrt{2}}{2}$, y must be equal to $\frac{\mathrm{\pi }}{4}$.

Hence, we get ${\sin }^{-1}\frac{\sqrt{2}}{2}=\frac{\mathrm{\pi }}{4}$.