Say I have a sum of two sinusoids like so:
$$ Acos(xt+\phi) + Bcos(yt+\delta) $$ How would I find the period? I know that for just one sinusoid the period would be:
$$ Acos(xt+\phi) $$ $$ T = 2\pi/x $$ It can't be as simple as just adding the two periods.
How can I tell if they're not periodic?
Beware that assuming there is a period to find is a big assumption. For example, suppose your two sinusoids are $\cos{2\pi t}$ and $\cos{\frac{2\pi t}{\sqrt{2}}}$. The periods of the two functions are $1$ and $\sqrt{2}$, respectively. If their sum was periodic, its period would be the least common multiple of $1$ and $\sqrt{2}$, which would allow us to write $\sqrt{2}$ as a ratio of integers (red flag!).
You need the sum-to-product identity: $$cos(\theta)+cos(\alpha)=2cos(\frac{\theta+\alpha}{2} )*cos(\frac{\theta-\alpha}{2} )$$
So your new period T is a "beat" signal or a sinusoid modulated by another (sort of the AM radio signals), where $$T_1=\frac{2\pi}{\frac{\theta+\alpha}{2}}=\frac{4\pi}{\theta+\alpha}$$
and $$T_2=\frac{2\pi}{\frac{\theta-\alpha}{2}}=\frac{4\pi}{\theta-\alpha}$$
Plot or check http://en.wikipedia.org/wiki/Beat_(acoustics) to grasp the periodicity of this.
Another important thing to consider is the ratio between A and B, (in AM radio this is called modulation index: see this link: http://en.wikipedia.org/wiki/File:Amplitude_Modulated_Wave-hm-64.svg), as the periodicity will be different when A>B than B>A.
