Question

how many periods of the function y = tan x are there between -4 π and π?

Explanation:

Step1: Recall the period of \( y = \tan x \)

The tangent function \( y=\tan x \) has a period of \( \pi \), which means the function repeats its values every \( \pi \) units.

Step2: Calculate the length of the interval

First, find the length of the interval from \( - 4\pi \) to \( \pi \). The length \( L \) is calculated as \( \pi-(-4\pi)=\pi + 4\pi=5\pi \).

Step3: Find the number of periods

To find the number of periods, divide the length of the interval by the period of the function. The period of \( \tan x \) is \( \pi \), so the number of periods \( n=\frac{5\pi}{\pi}=5 \).

Answer:

\( 5 \)