Question

In Problems 9–36, find the exact value of each expression.

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

Short answer

The value of the expression$\cos \left[{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\frac{1}{2}$.

Step-by-step solution

Step 1. Given information. 

The given expression is:

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

Let's take$\theta ={\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

$\sin \theta =\left(-\frac{\sqrt{3}}{2}\right)$

$-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$is the domain of the function.

Step 2. Find the value of θ

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

By using the unit circle, we can say that $\theta$must be equal to $\frac{5\mathrm{\pi }}{3}$, so that we can get $-\frac{\sqrt{3}}{2}$.

$$\begin{gathered} \sin \theta =\left(-\frac{\sqrt{3}}{2}\right) \\ {\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\frac{5\mathrm{\pi }}{3} \end{gathered}$$

Step 3. Find the exact value of the expression.

$$\begin{gathered} \cos \left[{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\cos \left(\frac{5\mathrm{\pi }}{3}\right) \\ =\frac{1}{2} \end{gathered}$$

Therefore,$\cos \left[{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\frac{1}{2}$.