How do I determine if the following function is periodic? 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?
 A: 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.
A: 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 $\cos(x+6\pi)=\cos(x+4\pi)=\cos(x+2\pi)=\cos(x)$.
A: 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?
A: 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.
