# sin x+sin 3x=0

$$\displaystyle{\sin{{x}}}+{\sin{{3}}}{x}={0}$$

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curwyrm

$$\displaystyle{\sin{{x}}}+{\sin{{3}}}{x}={2}\frac{{\sin{{\left({3}{x}+{x}\right)}}}}{{2}}\times\frac{{\cos{{\left({3}{x}-{x}\right)}}}}{{2}}={0}$$
$$\displaystyle{2}{\sin{{2}}}{x}{\cos{{x}}}={0}$$
$$\displaystyle{\sin{{2}}}{x}={0}$$ or $$\cos x=0$$
if $$\displaystyle{\sin{{2}}}{x}={0}$$, then
$$\displaystyle{2}{x}={0}{\quad\text{or}\quad}\pi{\quad\text{or}\quad}{2}\pi$$
$$\displaystyle{x}={0},\frac{\pi}{{2}},\pi$$
if $$\displaystyle{\cos{{x}}}={0}$$ then
$$\displaystyle{x}=\frac{\pi}{{2}},{3}\frac{\pi}{{2}}$$

content_user