Determining the period of $\sin 2x +\sin\frac{x}{2}$ without using the LCM of periods

Question

I tried to calculate period of function described as: $$y=\sin 2x +\sin\frac{x}{2}$$ but without using LCM of periods.

From definition of periodic function we have:

$$\begin{align} 0 &= \sin(2x+2T)+\sin\left(\frac{x+t}{2}\right)-\sin2x-\sin\left(\frac{x}{2}\right) \\[4pt] &=2\sin\left(\frac{5x+5T}{4}\right)\cos\left(\frac{x+T}{4}\right)-2\sin\left(\frac{5x}{4}\right)\cos\left(\frac{3x}{4}\right) \end{align}$$

I cannot do further more.

Answer 1

Let $$\sin2(x+T)+\sin\frac{x+T}{2}=\sin2x+\sin\frac{x}{2},$$ where $T>0$.

Thus, for $x=0$ we obtain: $$\sin2T+\sin\frac{T}{2}=0$$ and for $x=2\pi$ we obtain: $$\sin2T-\sin\frac{T}{2}=0,$$ which gives $$\sin2T=\sin\frac{T}{2}=0$$ or $$T=2\pi k,$$ where $k$ is a positive integer number.

We see that $k=1$ is not valid, but $k=2$ gives the answer: $$T=4\pi.$$