Chapter 5.3, Problem 39E

To determine

To find: All the solutions of the equation ${\csc }^{2}x-2=0$ in $\left[0,2\pi \right)$ algebraically. Use the table feature of a graphing utility to check the answers numerically.

Answer to Problem 39E

  $x=\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

Explanation of Solution

Given:

  ${\csc }^{2}x-2=0$

Calculation:

The given equation is ${\csc }^{2}x-2=0$

Simplify this equation

  $\begin{array}{l}{\csc }^{2}x=2\\ \csc x=\pm \sqrt{2}\end{array}$

The value of $\csc x$ is $\sqrt{2}$ is at $\frac{\pi }{4}$

  $\csc x$ is positive in first and second quadrant

And $\csc x$ is negative in third and fourth quadrant

Therefore, the solutions of the given equation in $\left[0,2\pi \right)$ are

  $\begin{array}{l}x=\frac{\pi }{4},\left(\pi -\frac{\pi }{4}\right),\left(\pi +\frac{\pi }{4}\right),\left(2\pi -\frac{\pi }{4}\right)\\ x=\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}\end{array}$

Now, using the table feature of graphing utility, the value for x are

  $x=\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

Hence, the solutions are verified.