I am having trouble calculating the fundamental period of functions that involve trigonometry. The exercise is to find the fundamental period of these expressions: $$\begin{align} f(x)&=\left(\sin 4x\right)^3 \\[4pt] g(x)&=\sin\frac{x}{2}+\sin\frac{x}{3} \end{align}$$
I am trying things like scratching graphs and writing $f(x)=f(x+T)$, but when I find a result I can't say it is the fundamental period.
On
In general for a sine wave $\sin{\omega t}$ the period is $T=|\frac{2\pi}{\omega}|.$ You can easily check this. If you have an even power of sine, see that now it is always positive, so the period is twice as before. For an odd power of sine, the period just remains $T=|\frac{2\pi}{\omega}|$, because it is just scaled vertically. So the period of $f(x)$ is just $T=|\frac{2\pi}{4}|=\frac{\pi}{2}$
It is easy to see that for $\sin{\frac x2}$ we have $T=4\pi$, and for $\sin{\frac x3}$ we have $T=6\pi$, so the question is what is the period of the sum of these functions ($g(x)$). In general if you have a sum of periodic functions, the total period will be the least common multiple, because this represents the minimum value for wich all of the functions cycled. Some (the ones with smaller periods) will have cycled more than others, but at the lcm, all functions will be reseted, ready for a fresh start.
Therefore, the period of $g(x)$ is $12\pi$.
One simple way is to check when the functions attain their maxima. $f(x)$ has maximum for $\sin(4x)=1$ or $x=\frac{1}{4}\left(\frac{\pi}{2}+2n\pi\right)$, so the period must be a multiple of $\pi/2$. You can easily see now that $f(x+\pi/2)=(\sin(4x+4\pi/2))^3=(\sin(4x+2\pi))^3=(\sin(4x))^3$.
For $g(x)$ I would look at the periodicity of each of the terms, and find the least common multiple. The other option is look when the function is zero, but remember to check multiples of that value, since the function has both positive and negative values.