Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The given expression is:

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

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

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

$0\le \theta \le \mathrm{\pi }$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 role="math" localid="1646415761203" ${\cos }^{-1}$.

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

$$\begin{gathered} \cos \theta =\frac{1}{2} \\ {\cos }^{-1}\frac{1}{2}=\frac{\mathrm{\pi }}{3} \end{gathered}$$

Step 3. Find the exact value of the expression.

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

By using the unit circle, we know that $\cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2}$. Then,

$$\begin{gathered} \sec \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{{\displaystyle \frac{1}{2}}} \\ \sec \left(\frac{\mathrm{\pi }}{3}\right)=2 \end{gathered}$$

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