Chapter 7.5, Problem 37E

To determine

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

Answer to Problem 37E

  $\left(\frac{7\pi }{6}\text{+2}\pi k\right)\text{and}\left(\frac{11\pi }{6}\text{+2}\pi k\right)$

Explanation of Solution

Given information:

  $\sin x=-\frac{1}{2}$

Calculation:

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

  $\begin{array}{c}\sin x=-\frac{1}{2}\\ x={\sin }^{-1}\frac{1}{2}\\ =\frac{7\pi }{6}\text{+2}\pi k\text{ and}\frac{11\pi }{6}\text{+2}\pi k\end{array}$

Thus the values of $x$ are $\left(\frac{7\pi }{6}\text{+2}\pi k\right)\text{and}\left(\frac{11\pi }{6}\text{+2}\pi k\right)$ .