Chapter 5.3, Problem 34E

To determine

To solve: the equation.

Answer to Problem 34E

  $\begin{array}{l}x=\frac{\pi }{3}+2n\pi \\ x=\frac{2\pi }{3}+2n\pi \\ x=\frac{4\pi }{3}+2n\pi \\ x=\frac{5\pi }{3}+2n\pi \end{array}$

Explanation of Solution

Given information:

Given information:

Given trigonometric equation

  $3{\csc }^{2}x=4$

Calculation:

Rewrite the trigonometric equation and isolating the trigonometric part on the left hand side.

  $\begin{array}{c}3{\csc }^{2}x=4\\ {\csc }^{2}x=\frac{4}{3}\\ \csc x=\pm \sqrt{\frac{4}{3}}\\ \csc x=\pm \frac{2}{\sqrt{3}}\end{array}$

Because $\csc x$ has a period of $2\pi$ to the obtained solution in $\left[0,2\pi \right)$ to obtain a general form of solution.

  $\begin{array}{l}x=\frac{\pi }{3}+2n\pi \\ x=\frac{2\pi }{3}+2n\pi \\ x=\frac{4\pi }{3}+2n\pi \\ x=\frac{5\pi }{3}+2n\pi \end{array}$

Where $n$ is an integer.