# How do you find the exact values of sin^-1(sqrt2/2)?

Sep 11, 2016

$\frac{\pi}{4}$

#### Explanation:

When you see either $\arcsin$ or ${\sin}^{-} 1$, you are looking for the ANGLE with a sin of the indicated value.

The range of $\arcsin$ or ${\sin}^{-} 1$ is between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$.
In other words, answers to this type of problem must fall between these values, which correspond to the 1st and 4th quadrant in the unit circle. And, this type of problem has only ONE answer.

Looking at the unit circle, you can see that the angle with a sin of positive $\frac{\sqrt{2}}{2}$ is $\frac{\pi}{4}$.

FYI, if the problem had been ${\sin}^{-} 1 \left(- \frac{\sqrt{2}}{2}\right)$, the answer would be $- \frac{\pi}{4}$, NOT $\frac{7 \pi}{4}$ because the answer must fall between

$- \frac{\pi}{2}$ and $\frac{\pi}{2}$.

$- \frac{\pi}{4}$ falls within this range but $\frac{7 \pi}{4}$ does not.