Question

if $y = \cos x$, how many periods will there be between $-4\pi$ and $2\pi$ ?

Explanation:

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

The period of the cosine function \( y=\cos x \) is \( 2\pi \), which means one full cycle (period) occurs over an interval of length \( 2\pi \).

Step2: Calculate the length of the interval

First, find the length of the interval from \( -4\pi \) to \( 2\pi \). The length \( L \) of an interval from \( a \) to \( b \) (where \( b > a \)) is given by \( L=b - a \). So, \( L = 2\pi-(-4\pi)=2\pi + 4\pi = 6\pi \).

Step3: Find the number of periods

To find the number of periods in this interval, divide the length of the interval by the length of one period. Let \( n \) be the number of periods. Then \( n=\frac{\text{Length of interval}}{\text{Length of one period}}=\frac{6\pi}{2\pi}=3 \).

Answer:

\( 3 \)