Question

True or False The period of \(y=\sin (x / 2)\) is \(\pi\) Justify your answer.

Short answer

The period of \(y=\sin (x / 2)\) is not \(\pi\). Thus, the statement in the exercise is False.

Step-by-step solution

Understand the Basic Period of the Sine Function

The basic period of the sine function \(\sin x\) is \(2\pi\). This means, a complete cycle of the curve of \(\sin x\) repeats every \(2\pi\). This can be observed from the unit circle or graph of \(\sin x\).

Determine How Coefficient Affects the Period

The coefficient of \(x\) in the sine function affects its period. Normally, if we have a function of the form \(\sin(bx)\), the period of the function becomes \(2\pi/b\). Here \(b\) is the coefficient of \(x\). This is because the changing \(x\) to \(bx\) effectively speeds up the rate at which we move around the unit circle, hence the function completes a cycle quicker.

Apply the Coefficient to Find the Period of Given Function

Applying this to our function \(y=\sin (x / 2)\), we note that \(b = 1/2\). Using the formula for period, we get \(2\pi/b = 2\pi/(1/2) = 4\pi\).

Compare the Obtained Period with \(\pi\)

We obtained that the period of the given function \(y=\sin (x / 2)\) is \(4\pi\). This is not equal to \(\pi\), which was supposed to be the period of the function according to the task.