Question

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

$sec\left({\cos }^{-1}\left(-\frac{3}{4}\right)\right)$

Short answer

The exact value of the given expression is $-\frac{4}{3}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$sec\left({\cos }^{-1}\left(-\frac{3}{4}\right)\right)$

Assume $\theta ={\cos }^{-1}\left(-\frac{3}{4}\right)$.

Using the inverse function,

$\cos \theta =-\frac{3}{4}$

We can also say,

$\sin \theta =\frac{opposite}{hypotenuse}$

Step 2. Rewrite the expression.

On rewriting the expression,

$$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{-3}{4}\right) \\ sec\theta =sec\left({\cos }^{-1}\left(\frac{-3}{4}\right)\right) \end{gathered}$$

Then,

$$\begin{gathered} sec\theta =\frac{1}{\cos \theta } \\ =\frac{1}{\left({\displaystyle \frac{-3}{4}}\right)} \\ =\left(-\frac{4}{3}\right) \end{gathered}$$

Therefore, $sec\left({\cos }^{-1}\left(-\frac{3}{4}\right)\right)=-\frac{4}{3}$.