Finding $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ periodicity

Question

I need to find the period of $\sin^3\frac{x}{2}\cdot\cos^7\frac{x}{3}$ but couldn't find a nice method how? any ideas?

Answer 1

Notice that: $$\sin\left(\frac{x}{2}\right) = \sin\left(\frac{x}{2} + 2k\pi\right) = \sin\left(\frac{x+4k\pi}{2}\right)$$

and

$$\cos\left(\frac{x}{3}\right) = \cos\left(\frac{x}{3} + 2h\pi\right) = \cos\left(\frac{x+6h\pi}{3}\right)$$

We need to find the minimum common period between $4k\pi$ and $6h\pi$ which is $12\pi$ (set $k=3$ and $h=2$).

Indeed:

$$\sin^3\left(\frac{x+12\pi}{2}\right)\cos^7\left(\frac{x+12\pi}{3}\right) = \sin^3\left(\frac{x}{2}+6\pi\right)\cos^7\left(\frac{x}{3}+4\pi\right) = \sin^3\left(\frac{x}{2}\right)\cos^7\left(\frac{x}{3}\right).$$

The period is $12\pi$.