the function is periodic but not a sinusoid. find the period graphically and sketch a graph showing one…

Explanation:

Step1: Recall period - formula for trigonometric functions

The period of $y = A\cos(Bx)$ is $T_1=\frac{2\pi}{|B|}$ and the period of $y = A\sin(Bx)$ is $T_2=\frac{2\pi}{|B|}$. For $y_1=\cos(2x)$, $B = 2$, so $T_1=\frac{2\pi}{2}=\pi$. For $y_2=- 2\sin(5x)$, $B = 5$, so $T_2=\frac{2\pi}{5}$.

Step2: Find the least - common multiple of the periods

We need to find the least - common multiple (in the sense of periods) of $\pi$ and $\frac{2\pi}{5}$. Let $m,n\in\mathbb{Z}$. We want to find the smallest positive $T$ such that $T = m\pi=n\frac{2\pi}{5}$. Cross - multiply to get $5m = 2n$. The smallest positive integers $m$ and $n$ that satisfy this equation are $m = 2$ and $n = 5$. Then $T = 2\pi$.

Answer:

$2\pi$