# Finding the period of $f(x) = \sin 2x + \cos 3x$

I want to find the period of the function $f(x) = \sin 2x + \cos 3x$. I tried to rewrite it using the double angle formula and addition formula for cosine. However, I did not obtain an easy function. Another idea I had was to calculate the zeros and find the difference between the zeros. But that is only applicable if the function oscillates around $y = 0$, right? My third approach was to calculate the extrema using calculus and from that derive the period. Does anybody have another approach? Thanks

• Hint: You know the period of $\sin 2x$ and of $\cos 3x$. You should compare them already and see if something can pop up. It should give you an idea of at least one period of the sum. May 21, 2015 at 13:30

Hint

The period of $\sin(2x)$ is $\pi$, and the period of $\cos(3x)$ is $2\pi/3$.

Can you find a point where both will be at the start of a new period?

• So the answer is $lcm(\pi,2\pi/3)=2\pi$? May 21, 2015 at 13:45
• Yes, that will be it May 21, 2015 at 17:43

In general, if $T$ is the period of a function $f(x)$ then the period of the function $f(ax)$ is $\frac{T}{a}$

In general, if two periodic functions $f_1(x)$ & $f_2(x)$ have periods $T_1$ & $T_2$ then the period of the function $g(x)=f_1(x)\pm f_2(x)$ is L.C.M. (Least common multiple) of $T_1$ & $T_2$.

As per your question, the periods of $\sin2x$ & $\cos3x$ are calculated as $\frac{2\pi}{2}=\pi$ & $\frac{2\pi}{3}$ respectively.

Hence the period of the function $f(x)=\sin2x+\cos3x$ is L.C.M. of $\pi$ & $\frac{2\pi}{3}$ which is given by generalized formula of L.C.M. of fractions $$=\frac{\text{L.C.M. (least common multiple) of numerators}}{\text{H.C.F. (highest common factor) of denominators}}=\frac{2\pi}{1}=2\pi$$ Hence, the required period is $2\pi$

$\sin(2x)=\sin(2(x+k\pi))$ and $\cos(3x)=\cos(3(x+l2\pi/3))$, then when $k=\dfrac{2l}3$, $3k=2l=6m$, the function repeats itself, and the period is at most $2\pi$.

Anyway, remains to prove that there is no shorter period.