Question

find the period of the sinusoidal function.
simplify any fractions.
period = \square

Explanation:

Step1: Identify peak positions

Find two consecutive peaks. For example, one peak at \( x = -7\pi \), next at \( x = -3\pi \), then at \( x = \pi \), \( x = 5\pi \), etc.

Step2: Calculate the period

The period is the difference between x - coordinates of consecutive peaks. Using \( x = -7\pi \) and \( x = -3\pi \): \( \text{Period} = -3\pi - (-7\pi) = 4\pi \). Check with another pair, e.g., \( x = -3\pi \) and \( x = \pi \): \( \pi - (-3\pi) = 4\pi \). Consistent.

Answer:

\( 4\pi \)