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
 A: In general, if $T$ is the period of a function $f(x)$ then the period of the function $f(ax)$ is $\frac{T}{a}$.
Suppose two periodic functions $f_1(x)$ and  $f_2(x)$ have periods $T_1$ and $T_2$. Then the period of the function $g(x)=f_1(x)\pm f_2(x)$ is LCM (least common multiple) of $T_1$ and $T_2$ (although, this certainly isn't true for all periodic functions, as explained inside this answer.)
In your question the periods of $\sin x$ and $\sin 3x$ are calculated as $\frac{2\pi}{1}=2\pi$ and $\frac{2\pi}{3}$ respectively. 
So the period of the function $f(x)=\sin x+\sin3x$ is the $\text{LCM}(2\pi,\frac{2\pi}{3})=2\pi$.
A: The period of $~\sin x~$ is $~2\pi~$ and  that of $~\sin 3x~$ is $~\frac{2\pi}{3}~$, because $~\sin\left(3\frac{2\pi}{3}\right)=\sin 2\pi~$.
Now given that $~f(x)=\sin 3x+\sin x~$
Let $~a~$be a period $~f(x)~$, then by the definition $~f(x+a)=f(x)~$.
Here $~f(x+a)=\sin(3x+3a)+\sin(x+a)~$
From $~\sin(3x+3a)~$, we have $~a=\frac{2\pi}{3}n,~~ n$ is integer, because its the same as adding period of $~\sin 3x, ~~n$ times.
Similarly, from $~\sin(x+a)~$, we have $~a=2\pi k~$, $k$ is integer.
$~a~$ is a multiple of $~\frac{2\pi}{3}~$ and $~2\pi~$, so the period is smallest positive multiple of $~\frac{2\pi}{3}~$ and $~2\pi~$ which is $~2\pi~$, because $~2\pi=2\pi*1 (\text{multiple of}~ 2\pi), 2\pi=\frac{2\pi}{3}*3 (\text{multiple of}~ \frac{2\pi}{3})~$.
The period of $~f(x)~$, $~a=2\pi~$
A: $f(x)=f(x+2\pi)$ 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 $2\pi$.
A: The period of $sin(x)$ is $2\pi$ and the period of $sin(3x)$ is $\displaystyle \frac{2\pi}{3}$. The period is thus $2\pi$.
