Question

Find the exact value of the expression. Express angles in radians.

$se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$

Short answer

The exact value of the given expression is$\frac{\mathrm{\pi }}{6}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$

Assume $\theta =se{c}^{-1}\left(\frac{2}{\sqrt{3}}\right)$.

As $y={\sin }^{-1}x$means $x=\sin y$, where,

$-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

Rewrite the function,

$sec\theta =\frac{2}{\sqrt{3}}$

Step 2. Use the reciprocal identity.

Using the reciprocal identity, $sec\theta =\frac{1}{\cos \theta }$,

Then,

$$\begin{gathered} \frac{1}{\cos \theta }=\frac{1}{{\displaystyle \frac{2}{\sqrt{3}}}} \\ \cos \theta =\frac{\sqrt{3}}{2} \end{gathered}$$

We know $\cos \frac{\mathrm{\pi }}{6}=\frac{\sqrt{3}}{2}$and as $\frac{\mathrm{\pi }}{6}$is in the interval $0\le \theta \le \mathrm{\pi }$.

Then, we can say,

localid="1646501788181" $$\begin{gathered} \cos \theta =\cos \frac{\mathrm{\pi }}{6} \\ \theta =\frac{\mathrm{\pi }}{6} \end{gathered}$$