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I would like to prove that the function

\begin{equation} \sin(2x)+\cos(3x) \end{equation}

is periodic and calculate its period. I tried to replace $x+T$ instead of $x$, and I got:

\begin{equation} \sin(2x)\cos(2T)+\cos(2x)\sin(2T)+\cos(3x)\cos(3T)-\sin(3x)\sin(3T) \end{equation}

From this point I do not know how to continue. Any suggestions, please?

Thank you very much

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which should be equal to $\sin(2x)+\cos(3x)$ , finally you will get some equations that $T$ has to satisfy . – Theorem Nov 20 '12 at 19:58
$\sin(2x)$ has period $\pi$, and $\cos(3x)$ has period $2\pi/3$. $2\pi$ is a least common natural multiple of the two. – Arthur Nov 20 '12 at 20:00
up vote 4 down vote accepted

From where you left, you need

\begin{equation} \sin(2x)\cos(2T)+\cos(2x)\sin(2T)+\cos(3x)\cos(3T)-\sin(3x)\sin(3T)=\sin(2x)+\cos(3x) \end{equation} for all $x$. As the functions $\sin(2x)$, $\sin(3x)$, $\cos(2x)$, $\cos(3x)$ are linearly independent, this forces $$ \cos(2T)=1,\ \sin(2T)=0,\ \cos(3T)=1,\ \sin(3T)=0. $$ The equalities $\sin(2T)=\sin(3T)=0$ imply that $T=k\pi$ for some $k\in\mathbb N$ (positive, because $T$ is a period). The $\cos(2k\pi)=1$ for all $k$, but $\cos(3k\pi)=1$, requires $k$ to be even. So the smallest possible value is $T=2\pi$.

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Why is $\sin(2T)=\sin(3T)$ ? – GinKin Dec 13 '13 at 17:26
They both need to be zero for the first equality above to hold. – Martin Argerami Dec 13 '13 at 21:04

It is clear that $2\pi$ is a period of $\sin 2x +\cos 3x$. We show that $2\pi$ is the smallest (positive) period.

Let $\alpha$ be a period of our function. Then for all $x$, $$\sin 2x+\cos 3x=\sin 2(x+\alpha) +\cos 3(x+\alpha).\tag{$1$}$$ Differentiate twice, and change signs. (It would be enough to differentiate once, but twice is neater.) We get $$4\sin 2x+9\cos 3x= 4\sin 2(x+\alpha) +9\cos 3(x+\alpha).\tag{$2$}$$ Use Equations $(1)$ and $(2)$ to eliminate the $\cos$ terms. We get $$5\sin 2x =5\sin 2(x+\alpha).$$ Thus $\alpha$ is a period of $\sin 2x$. Similarly, $\alpha$ is a period of $\cos 3x$. But the smallest common positive period of $\sin 2x$ and $\cos 3x$ $\alpha$ is $2\pi$.

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Show that this function returns the same values for $x$ and $x+2\pi$ by direct substitution

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You nee the argument of @Arthur to show that this is the period. – abatkai Nov 20 '12 at 20:12
Yes. I just thought that he only wants to show that the function is periodic – Amr Nov 20 '12 at 20:15

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