How do I determine if the following function is periodic?

Think about this problem intuitively. Consider a drag racer where the hind wheels have radii 3X the radii of the smaller front wheels. If we mark the wheels (on one side of the car) where they initially touch the ground with yellow chalk, how often will the yellow chalk on both wheels be touching the ground simultaneously?

Hint: A function $f$ is periodic with a period $T$ if $f(x) = f(x + T)$. $T$ should be non-zero.

Can you spot a common period for both summands?

Try to plot the function to get an idea what is going on.

Hint: $$ \begin{align} \cos(3(x+2\pi))+\sin(x+2\pi) &=\cos(3x+6\pi)+\sin(x+2\pi)\\ ]\end{align} $$ Now apply the fact that $\sin(x+2\pi)=\sin(x)$ and $\cos(x+2\pi)=\cos(x)$.

Note that the last identity says that $\cos(x+6\pi)=\cos(x+4\pi)=\cos(x+2\pi)=\cos(x)$.