How to know if a function is periodic without graphing it ?

pi * sin(pi*t/2) + 1.8cos(7*pi*t/5)

A function is periodic if there is some constant such that for all in the domain of . Then is the "period" of .

Example:

If , then we have , and so is periodic with period .

It gets a bit more complicated for a function like yours. We're looking for such that



Expanding on the left, you have



and



It follows that the following must be satisfied:



The first two equations are satisfied whenever , or more generally, when and (i.e. any multiple of 4).

The second two are satisfied whenever , and more generally when with (any multiple of 10/7).

It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.

Let's verify:





More generally, it can be shown that



is periodic with period .

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