What is the period of f(t)=sin( t / 32 )+ cos( (t)/64 ) ?

May 29, 2016

Both $\sin$ and $\cos$ are periodic with period $2 \pi$.
Then, for example, $\sin \left(t\right) + \cos \left(t\right)$ is automatically periodic of $2 \pi$ because if we substitute $t = 2 \pi$ both functions return on the initial value and so does their sum.

Now the period of the function $\sin \left(\frac{t}{32}\right)$ is $64 \pi$ because when $t = 64 \pi$ we have $\sin \left(2 \pi\right)$ that is equal to $\sin \left(0\right)$ and then the function restarts.

Applying the same concept $\cos \left(\frac{t}{64}\right)$ has the period $128 \pi$.

This means that if we take the sum, when we arrive to $64 \pi$ the $\sin$ did a full turn but the $\cos$ is still not repeating. When we are at $128 \pi$ the $\sin$ did two full turns ($4 \pi$) and the $\cos$ did its full period. So both functions are again to zero and the sum will restart the next cycle.

We are lucky that 128 is exactly the double of 64 so one period of the $\cos$ correspond to exactly two periods of $\sin$. If this is not true we have to search the least common multiple of both periods to have a period that is valid for both functions. In fact $128$ is the LCM of $128$ and $64$.
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