Chapter 7.5, Problem 42E

To determine

To find: all the real values of $x$ by solving the equation.

Answer to Problem 42E

  $\left(\frac{\pi }{3}+2\pi k\right)\text{ and }\left(\frac{5\pi }{3}+2\pi k\right)$

Explanation of Solution

Given information:

  $2{\tan }^{2}x-3\sec x=0$

Calculation:

The value of $x$ for the given equation can be evaluated as follows:

  $\begin{array}{c}2{\tan }^{2}x-3\sec x=0\end{array}$

  $\begin{array}{c}2\left({\sec }^{2}x-1\right)-3\sec x=0\left[\text{Using Pythagorean identity of tangent}\right]\end{array}$

  $\begin{array}{c}2{\sec }^{2}x-2-3\sec x=0\end{array}$

  $\begin{array}{c}\left(2\sec x+1\right)\left(\sec x-2\right)=0\left[\text{Factors}\right]\end{array}$

  $\begin{array}{c}2\sec x+1=0\text{ or }\sec x-2=0\end{array}$

  $\begin{array}{c}\sec x=-\frac{1}{2}\text{ or }\sec x=2\end{array}$

  $\begin{array}{c}\frac{1}{\cos x}=-\frac{1}{2}\text{ or }\frac{1}{\cos x}=2\end{array}$

  $\begin{array}{c}\cos x=-2\text{ or }\cos x=\frac{1}{2}\left[\because \cos x\ne -2\right]\end{array}$

  $\begin{array}{c}x=\left(\frac{\pi }{3}+2\pi k\right)\text{ and }\left(\frac{5\pi }{3}+2\pi k\right)\left[\text{Here }k\text{ is an integer}\right]\end{array}$

Thus the values of $x$ are $\left(\frac{\pi }{3}+2\pi k\right)\text{ and }\left(\frac{5\pi }{3}+2\pi k\right)$ .