Chapter 6.4, Problem 96AYU

To determine

To calculate: four of the intersection points out of the infinite intersection points where the pair of function intersects.

Answer to Problem 96AYU

Solution: $\left(\frac{\pi }{6},\frac{1}{2}\right),\left(\frac{5\pi }{6},\frac{1}{2}\right),\left(\frac{13\pi }{6},\frac{1}{2}\right),\left(\frac{17\pi }{6},\frac{1}{2}\right)$

Explanation of Solution

  $\begin{array}{l}y=\sin x\\ y=\frac{1}{2}\end{array}$

Solve the equation:

  $\sin x=\frac{1}{2}$

Sin function has period $2\pi$ . In $[0,2\pi )$ , equation has two solutions

  $x_1=\frac{\pi }{6},x_2=\frac{5\pi }{6}$

The general solution of the equation is

  $x=\frac{\pi }{6}+2k\pi orx=\frac{5\pi }{6}+2k\pi$ , $k$ integer

  $\begin{array}{l}{x}_3=\frac{\pi }{6}+2\pi =\frac{13\pi }{6}\\ x_4=\frac{5\pi }{6}+2\pi =\frac{17\pi }{6}\end{array}$

  $\left(\frac{\pi }{6},\frac{1}{2}\right),\left(\frac{5\pi }{6},\frac{1}{2}\right),\left(\frac{13\pi }{6},\frac{1}{2}\right),\left(\frac{17\pi }{6},\frac{1}{2}\right)$