Question

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

sin^-1 (- 1)

Short answer

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

Step-by-step solution

Step 1. Given Information:

Given expression: sin^-1 (- 1)

We want to find the exact value of given expression.

Step 2. Solution:

The given expression can be rewritten as:

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

Hence we get sin (y) = - 1

Since we have sin^-1 function initially, the bounds on the sin function are going to be in the interval localid="1647536843358" $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$ which represents the range of sin^-1 function.

Using the unit circle, we know that in order to get (- 1), y must be equal to $-\frac{\mathrm{\pi }}{2}$.

Hence, we getlocalid="1647536511185" $si{n}^{-1}(-1)=-\frac{\mathrm{\pi }}{2}$