Question

12 mark for review
the functions $f$ and $g$ are given by $f(\theta)=2\cos\theta$ and $g(\theta)=2\cos(b\theta)$, where $b$ is a constant. if the period of $g$ is half the period of $f$, then solving which of the following equations will give the correct value of $b$
a. $\frac{2\pi}{b}=\frac{1}{2}$
b. $\frac{2\pi}{b}=\frac{1}{2}(2\pi)$
c. $\frac{2\pi}{b}=2\pi$
d. $\frac{2\pi}{b}=2(2\pi)$

Explanation:

Step1: Find period of $f(\theta)$

The period of $\cos\theta$ is $2\pi$, so period of $f(\theta)=2\cos\theta$ is $2\pi$.

Step2: Find period of $g(\theta)$

For $g(\theta)=2\cos(b\theta)$, period is $\frac{2\pi}{|b|}$. Since period is positive, we use $\frac{2\pi}{b}$.

Step3: Set up period relationship

Given period of $g$ = $\frac{1}{2}$ period of $f$. Substitute periods:
$\frac{2\pi}{b} = \frac{1}{2}(2\pi)$

Answer:

B. $\frac{2\pi}{b} = \frac{1}{2}(2\pi)$