What is the period of the $f(x)=\sin x +\sin3x$? 
What is the period of the $f(x)=\sin x +\sin3x$?

$f(x)=\sin x+\sin 3x=2\frac{3x+x}{2}\cos\frac{x-3x}{2}=2\sin2x\cos x=4\sin x\cos^2x\\f(x+T)=4\sin(x+T)\cos^2(x+T)=4\sin x\cos^2 x$
how can I deduct this I have no idea

$f(x)=f(x+2\pi)$ so it has a period which is a factor of $2\pi$.
$df/dx=4$ only when $x$ is a multiple of $2\pi$ so any period is a multiple of $2\pi$.
So the period is $2\pi$.
