Chapter 14.2, Problem 44PPSE

Solution

a.

To write: The complete solution for $\sin \theta \ge 0.5$ .

  $\left[2n\pi +\frac{\pi }{6},2n\pi +\frac{5\pi }{6}\right]$

Given information:

A portion of the graphs of $y=\sin \theta$ and $y=0.5$ .

Calculation:

In the given portion, the graph of $y=\sin \theta$ intersects with the graph of $y=0.5$ at the points $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$ .

So, the solution of the inequality $\sin \theta \ge 0.5$ for the given portion of the graph is $\left[\frac{\pi }{6},\frac{5\pi }{6}\right]$ .

Since the period of $y=\sin \theta$ is $2\pi$ , the complete solution of $\sin \theta \ge 0.5$ is $\left[2\pi n+\frac{\pi }{6},2\pi n+\frac{5\pi }{6}\right]$ or $\left[2n\pi +\frac{\pi }{6},2n\pi +\frac{5\pi }{6}\right]$ .

b.

To write: The complete solution for $\sin \theta \le 0.5$ .

  $\left[2n\pi ,2n\pi +\frac{\pi }{6}\right]\cup \left[2n\pi +\frac{5\pi }{6},2n\pi +2\pi \right]$

Given information:

A portion of the graphs of $y=\sin \theta$ and $y=0.5$ .

Calculation:

In the given portion, the graph of $y=\sin \theta$ intersects with the graph of $y=0.5$ at the points $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$ .

So, the solution of the inequality $\sin \theta \le 0.5$ for the given portion of the graph is $\left[0,\frac{\pi }{6}\right]\cup \left[\frac{5\pi }{6},2\pi \right]$ .

Since the period of $y=\sin \theta$ is $2\pi$ , the complete solution of $\sin \theta \le 0.5$ is $\left[2\pi n+0,2\pi n+\frac{\pi }{6}\right]\cup \left[2\pi n+\frac{5\pi }{6},2\pi n+2\pi \right]$ or $\left[2n\pi ,2n\pi +\frac{\pi }{6}\right]\cup \left[2n\pi +\frac{5\pi }{6},2n\pi +2\pi \right]$ .

c.

To explain: The process of solving inequalities involving trigonometric functions.

Given information:

A portion of the graphs of $y=\sin \theta$ and $y=0.5$ .

Calculation:

To solve the inequalities involving trigonometric functions, first graph the inequalities on the same coordinate axes, then observe the points of intersections of the two graphs.

Then, the solution of the inequality will be the values of $x$ for which one function lies above or below the other function depending on the sign of inequality.