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A question in my textbook asks me to determine if the function $f(x)=\cos(3x)+\sin(x)$ is periodic. I do not believe this to be the case as the arguments of sine and cosine are different and as such they cannot be written as a sinusoid. How can I determine for certain if the function is periodic or not?

Here's my attempt at answering my own question, please critique where possible.

A function is periodic if $f(x+p)=f(x) \therefore \cos(3(x+p)) + \sin(x+p) = \cos (3x) + \sin(x)$

Letting $x = 0$:

$\cos(0)+\sin(0)=\cos(3p)+\sin(p)$

$1=\cos(3p)+\sin(p)$

Where do I go from here? I know that this would hold when: sine is 0 and cosine is 1 OR when cosine is 0 and sine is 1. However how do I ascertain for which value of P this occurs?

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    $\begingroup$ Try to simplify $f(x+2\pi)$ using the usual rules for $\sin$ and $\cos$ $\endgroup$ Commented Nov 3, 2014 at 13:50
  • $\begingroup$ @seeker, See math.stackexchange.com/questions/897987/… $\endgroup$ Commented Nov 3, 2014 at 18:26
  • $\begingroup$ Sketch the graph. It has two frequencies, two periods coming together. $\endgroup$
    – Narasimham
    Commented Aug 26, 2015 at 15:22

4 Answers 4

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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?enter image description here

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  • $\begingroup$ Every time they complete a full revolution? $\endgroup$
    – seeker
    Commented Nov 3, 2014 at 15:09
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    $\begingroup$ nice illustration btw $\endgroup$
    – seeker
    Commented Nov 3, 2014 at 15:24
  • $\begingroup$ BTW bl.ocks.org/jinroh/7524988 comes close to this idea, use settings "Square" and "4". Only that the small circle (~ cos term) has as much smaller radius than the big one (~ sin term) and the angle offset of the small one (phase) seems off by a quarter revolution. $\endgroup$
    – mvw
    Commented Nov 3, 2014 at 16:09
  • $\begingroup$ @mvw very nice link, ty $\endgroup$
    – John Joy
    Commented Nov 3, 2014 at 16:42
  • $\begingroup$ @seeker clearly every time the big wheel completes 1 revolution, the small wheel completes 3 of them. So the period should be $3(2\pi)=6\pi$ $\endgroup$
    – John Joy
    Commented Nov 3, 2014 at 16:46
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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.

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  • $\begingroup$ $\frac{3\pi}{4}$ radians? Is it also possible that you could suggest improvements to my answer? $\endgroup$
    – seeker
    Commented Nov 3, 2014 at 15:34
  • $\begingroup$ @seeker You made a mistake at the beginning: $f(x+p) = \cos(3(x+p))+\sin(x+p)$ $\endgroup$
    – mvw
    Commented Nov 3, 2014 at 15:50
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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 $\cos(x+6\pi)=\cos(x+4\pi)=\cos(x+2\pi)=\cos(x)$.

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To check if a function repeats itself with respect to time i.e after a fixed interval of time.

So we just have to interpret when the function is going to repeat.

Sine and cosine repeat at multiples $2\pi$.

$\cos3x+\sin x$, after $2\pi$ period of time $\cos3(x+2\pi)+\sin(x+2\pi)$ Which equal to $\cos3x+\sin x$ i.e the original function.

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