the period of a function is $4pi$. how many cycles of the function occur in a horizontal length of $12pi$?

Explanation:

Step1: Recall cycle - period relationship

The number of cycles $n$ of a periodic function in a given length $L$ is found by dividing the length $L$ by the period $T$ of the function, i.e., $n=\frac{L}{T}$.

Step2: Identify values of $L$ and $T$

We are given that $L = 12\pi$ and $T=4\pi$.

Step3: Calculate the number of cycles

Substitute the values into the formula: $n=\frac{12\pi}{4\pi}$. The $\pi$ terms cancel out, and $\frac{12}{4}=3$.

Answer:

3