Chapter 5, Problem 41RE

To determine

To find all the solutions of the given equation.

Answer to Problem 41RE

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

Explanation of Solution

Given information:

Given trigonometric equation

  $\sin x=\sqrt{3}-\sin x$

Calculation:

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

  $\begin{array}{c}\sin x=\sqrt{3}-\sin x\\ \sin x+\sin x=\sqrt{3}\\ 2\sin x=\sqrt{3}\\ \sin x=\frac{\sqrt{3}}{2}\end{array}$

Because $\sin x$ has a period of $2\pi$ , so we first find the solution in $\left[0,2\pi \right)$ .

These solutions are $x=\frac{\pi }{3}$ and $x=\frac{2\pi }{3}$ .

Finally add multiples of $2\pi$ to the obtained solutions in $\left[0,2\pi \right)$ to get the general form of solution.

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

Where $n$ is an integer.