# What is the period of sin(3*x)+ sin(x/(2))?

Jul 17, 2016

The Prin. Prd. of the given fun. is $4 \pi$.

#### Explanation:

Let $f \left(x\right) = \sin 3 x + \sin \left(\frac{x}{2}\right) = g \left(x\right) + h \left(x\right)$, say.

We know that the Principal Period of $\sin$ fun. is $2 \pi$. This

means that, $\forall \theta , \sin \left(\theta + 2 \pi\right) = \sin \theta$

$\Rightarrow \sin 3 x = \sin \left(3 x + 2 \pi\right) = \sin \left(3 \left(x + 2 \frac{\pi}{3}\right)\right)$

$\Rightarrow g \left(x\right) = g \left(x + 2 \frac{\pi}{3}\right)$.

Hence, the Prin. Prd. of the fun. $g$ is $2 \frac{\pi}{3} = {p}_{1}$, say.

On the same lines, we can show that, the Prin. Prd. of the fun $h$ is

$\frac{2 \pi}{\frac{1}{2}} = 4 \pi = {p}_{2}$, say.

It should be noted here that, for a fun. $F = G + H$, where,

$G \mathmr{and} H$ are periodic funs. with Prin. Prds. P_1 & P_2, resp.,

it is not at all necessary that the fun. $F$ be periodic .

However, $F$ will be so, with Prin. Prd. $p$, if we can find,

$l , m \in \mathbb{N}$, such that, $l \cdot {P}_{1} = m \cdot {P}_{2} = p$.

So, let us suppose that, in our case, for some $l , m \in \mathbb{N} ,$

$l \cdot {p}_{1} = m \cdot {p}_{2} = p \ldots \ldots \ldots \ldots . \left(1\right)$

$\Rightarrow l \cdot \frac{2 \pi}{3} = m \cdot 4 \pi \Rightarrow l = 6 m$

So, by taking, $l = 6 , \mathmr{and} m = 1$, we have, from $\left(1\right)$,

$6 \cdot \left(2 \frac{\pi}{3}\right) = 1 \cdot \left(4 \pi\right) = p = 4 \pi$

Hence, the Prin. Prd. of the given fun. is $4 \pi$.