Question

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

$\sec \left({\tan }^{-1}\sqrt{3}\right)$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given expression is:

$\sec \left({\tan }^{-1}\sqrt{3}\right)$

Let's take $\theta ={\tan }^{-1}\sqrt{3}$,

$\tan \theta =\sqrt{3}$

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

Step 2. Find the value of θ.

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

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

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

Step 3. Find the exact value of the expression.

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

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

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

Therefore,$\sec \left({\tan }^{-1}\sqrt{3}\right)=2$.