32. If the period of $f(x)=\sin (k x)$, where $k \neq 0$, is $A$ and the period of $g(x)=|\sin (k x)|$, where $k \neq 0$, is B, then
a) $A=2 \pi B$
b) $A=\frac{1}{2} B$
c) $A=2 B$
d) $A=\pi B$
e) $A=B$

Step 1 ：First, we find the period of $f(x)=\sin(kx)$. The graph of $f(x)$ passes through one full period as $kx$ ranges from $0$ to $2\pi$, which means $x$ ranges from $0$ to $\frac{2\pi}{k}$. So, the period of $f(x)$ is $A=\frac{2\pi}{k}$.

Step 2 ：Next, we find the period of $g(x)=|\sin(kx)|$. The graph of $g(x)$ passes through one full period as $kx$ ranges from $0$ to $\pi$, which means $x$ ranges from $0$ to $\frac{\pi}{k}$. So, the period of $g(x)$ is $B=\frac{\pi}{k}$.

Step 3 ：Finally, we compare the periods of $f(x)$ and $g(x)$. We have $A=\frac{2\pi}{k}$ and $B=\frac{\pi}{k}$. Dividing $A$ by $B$, we get $\frac{A}{B}=\frac{\frac{2\pi}{k}}{\frac{\pi}{k}}=2$.

Step 4 ：Thus, $A=2B$, which corresponds to option (c). So, the final answer is \(\boxed{\text{(c)}}\)