Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The given expression is:

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

Let's take localid="1646405774219" $\theta ={\cos }^{-1}\frac{1}{2}$,

$\cos \theta =\frac{1}{2}$

$0\le \theta \le \mathrm{\pi }$is the domain of the function.

Step 2. Find the value of θ.

Initially, we have ${\cos }^{-1}$and the bounds of cosfunction are going to be in the interval $\left[0,\mathrm{\pi }\right]$. It represents the range of ${\cos }^{-1}$.

By using the unit circle, we can say that$\theta$ must be equal to $\frac{\mathrm{\pi }}{3}$, so that we can get $\frac{1}{2}$.

$$\begin{gathered} \cos \theta =\frac{1}{2} \\ {\cos }^{-1}\frac{1}{2}=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Step 3. Find the exact value of the expression.

$\sin \left(\frac{\mathrm{\pi }}{3}\right)=\frac{\sqrt{3}}{2}$

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