A non periodic function made of two periodic functions

Algebra Level 3

I have two functions $f(x) = \cos(Ax)$ and $g(x) = \sin(Bx)$ . Let $h(x) = f(x) + g(x)$

If $h(x)$ is NOT periodic, what could be a possible value of $A + B$ ?

4 h(x) is always periodic pi + 1 e + pi Any of these 2e

1 solution

Timothy Cao
Mar 6, 2018

Brief explanation:

A summation of two periodic functions is not periodic if the two individual periods can never become equivalent after multiplying each by any non-zero integer.

i.e. in this case: $A * n = B * m$ (where n and m are integers) If this is not possible for a given A and B, then this is not periodic.

Each of the numbers in the question could be split into two different irrational numbers that are not multiples of each other. This will mean the periods will never sync up, and therefore the sum will not be periodic.

e.g. Looking at 4:

$4 = (2+\sqrt {2}) + (2-\sqrt {2})$ where 4 is split into two irrational numbers who do not share a finite LCM.

if $A = (2+\sqrt {2})$ and $B = (2-\sqrt {2})$ , then $h(x)$ is not periodic

For more on this topic: Refer to RedpenBlackpen's amazing video: https://www.youtube.com/watch?v=2K7_qyH8h5Y

A non periodic function made of two periodic functions

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