Chapter 14, Problem 19CLR

To determine

To solve: The equation $2{\csc }^{2}x+3\csc x=2$ for $x$ .

Answer to Problem 19CLR

The values of $x$ are $x=\frac{7\pi }{6},\frac{11\pi }{6}$ .

Explanation of Solution

Given:

The equation is $2{\csc }^{2}x+3\csc x=2$ and $0\le \theta <2\pi$ .

Calculation:

Simplify the equation as follows.

  $\begin{array}{c}2{\csc }^{2}x+3\csc x=2\\ 2{\csc }^{2}x+3\csc x-2=0\\ \left(2\csc x-1\right)\left(\csc x+2\right)=0\end{array}$

Set each factor to zero.

  $\begin{array}{c}2\csc x-1=0\\ \csc x=\frac{1}{2}\\ x=\text{not}\text{defined}\end{array}$

And,

  $\begin{array}{c}\csc x+2=0\\ \csc x=-2\\ x={\csc }^{-1}\left(-2\right)\\ =\frac{7\pi }{6},\frac{11\pi }{6}\end{array}$

Therefore, the values of $x$ are $x=\frac{7\pi }{6},\frac{11\pi }{6}$ .