Chapter 7.4, Problem 38E

To determine

To find: The solution of the equation ${\csc }^2\theta -4=0$ .

Answer to Problem 38E

The solutions of the equation ${\csc }^2\theta -4=0$ are $\theta =\frac{\pi }{6}+k\pi$ and $\theta =\frac{5\pi }{6}+k\pi$ .

Explanation of Solution

Given information:

The given equation is ${\csc }^2\theta -4=0$ .

Calculation:

Simplify the equation

  $\begin{array}{c}{\csc }^2\theta -4=0\\ {\csc }^2\theta =4\\ \csc \theta =\pm 2\end{array}$

As cosecant has period $2\pi$ . So first find the solutions in $\left[0,2\pi \right)$ .

Therefore, the solution to the equation in $\left[0,2\pi \right)$ are

  $\begin{array}{c}\theta ={\csc }^{-1}\left(2\right)\\ =\frac{\pi }{6},\frac{5\pi }{6}\end{array}$

And

  $\begin{array}{c}\theta ={\csc }^{-1}\left(-2\right)\\ =\frac{7\pi }{6},\frac{11\pi }{6}\end{array}$

As cosecant repeats its values after every $2\pi$ units.

So, the general solution of the equation is given by

  $\theta =\frac{\pi }{6}+2k\pi ,\frac{5\pi }{6}+2k\pi ,\frac{7\pi }{6}+2k\pi ,\frac{11\pi }{6}+2k\pi$

Therefore, the solutions of the equation ${\csc }^2\theta -4=0$ are $\theta =\frac{\pi }{6}+k\pi$ and $\theta =\frac{5\pi }{6}+k\pi$ .