Q. 23 - Precalculus Enhanced with Graphing Utilities

Question

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sec \left(\frac{7\pi }{4}\right)$

Step-by-Step Solution

Answer

The exact value is $\sqrt{2}$.

Step 1. Given information.

We have:

$\sec \left(\frac{7\pi }{4}\right)$

Step 2. Find the exact value of trigonometric function.

The function $\sec \left(\frac{7\pi }{4}\right)$.

$\sec \left(\frac{7\pi }{4}\right)=\frac{1}{\cos \left(\frac{7\pi }{4}\right)}$

Formula used:

$\cos \left(x\right)=\sin \left(\frac{\pi }{2}-x\right)$

$$\begin{gathered} =\frac{1}{\sin \left(\frac{\pi }{2}-\frac{7\pi }{4}\right)} \\ =\frac{1}{\sin \left(-\frac{5\pi }{4}\right)} \\ =-\frac{1}{\sin \left(\frac{5\pi }{4}\right)} \\ =-\frac{1}{-\frac{\sqrt{2}}{2}} \\ =\sqrt{2} \end{gathered}$$