Chapter 4.7, Problem 70E

To determine

To find: The exact value of the expression $\sec \left(\arctan \left(-\frac{3}{5}\right)\right)$ . Use a graphing utility to verify the result.

Answer to Problem 70E

  $\sec \left(\arctan \left(-\frac{3}{5}\right)\right)=-\frac{\sqrt{34}}{5}$

Explanation of Solution

Given:$\sec \left(\arctan \left(-\frac{3}{5}\right)\right)$

Calculation:

The given expression is $\sec \left(\arctan \left(-\frac{3}{5}\right)\right)$

This expression can be rewritten as

  $\begin{array}{l}\sec \left(-\arctan \left(\frac{3}{5}\right)\right)\\ =-\sec \left(\arctan \left(\frac{3}{5}\right)\right)\end{array}$

Let $\arctan \left(\frac{3}{5}\right)=y$ , then the given expression becomes $-\sec y$

Since, $\arctan \left(\frac{3}{5}\right)=y\Rightarrow \tan y=\frac{3}{5}$

Using this information, make a right triangle as shown below

  

Using Pythagoras Theorem,

  $\begin{array}{c}AC=\sqrt{3^2+5^2}\\ =\sqrt{9+25}\\ =\sqrt{34}\end{array}$

From the right triangle,

  $\begin{array}{c}\sec y=\frac{AC}{BC}\\ =\frac{\sqrt{34}}{5}\end{array}$

Therefore,

  $\sec \left(\arctan \left(-\frac{3}{5}\right)\right)=-\frac{\sqrt{34}}{5}$

Now, using the graphing utility, the value of the expression is $-\frac{\sqrt{34}}{5}$

Since, the calculated value is equal to the value that is found using a graphing utility.

Hence, the calculated value is true and this verified the result.