How to prove that $z(t) = z(t + T_0) \forall \theta$ for a periodic function

Question

i have the following question about a proof:

$z(t) = sin(4\pi t )+2cos(9\pi t+\theta)$

show that $z(t) = z(t + T_0) \forall \theta$

i dont know how to prove it. I know you should use $cos(9\pi t+\theta) = cos(9\pi t)cos(t\theta)-sin(9\pi t)sin(\theta)$

but from here i dont know what to do.

Answer 1

Some hints:

The function $t\mapsto\sin(4\pi t)$ has a certain period $T_1$, and the function $t\mapsto\cos(9\pi t+\theta)$ has a certain period $T_2$, whereby the $T_2$ does not depend on $\theta$. The ${\rm lcm}(T_1,T_2)$ is then a period of $t\mapsto z(t)$.