Chapter 0.6, Problem 50E

To determine

To state: The given statement is true of false.

Answer to Problem 50E

The given statement is false.

Explanation of Solution

Given information: The given statement is “The period of $y=\sin \left(\frac{x}{2}\right)$ is $\pi$ .

Consider the equation of general sine function.

  $f\left(x\right)=A\sin \left(B\left(x-C\right)\right)+D$

Here, amplitude is $\left|A\right|$ , period is $\left|\frac{2\pi }{B}\right|$ , $C$ is the horizontal shift, and $D$ is the vertical shift.

The given equation is $y=\sin \left(\frac{x}{2}\right)$ .

Compare the given equation with the general equation to get $B=\frac{1}{2}$ .

Substitute $\frac{1}{2}$ for $B$ in the formula of period.

  $\begin{array}{c}\left|\frac{2\pi }{B}\right|=\frac{2\pi }{\left|\frac{1}{2}\right|}\\ =4\pi \end{array}$

The period of the equation $y=\sin \left(\frac{x}{2}\right)$ is $4\pi$ , not $\pi$ .

Thus, the given statement is false.