# What is the period of f(t)=sin( t / 30 )+ cos( (t)/ 42) ?

Aug 1, 2018

#### Answer:

The period is $T = 420 \pi$

#### Explanation:

The period $T$ of a periodic function $f \left(x\right)$ is given by

$f \left(x\right) = f \left(x + T\right)$

Here,

$f \left(t\right) = \sin \left(\frac{t}{30}\right) + \cos \left(\frac{t}{42}\right)$

Therefore,

$f \left(t + T\right) = \sin \left(\frac{1}{30} \left(t + T\right)\right) + \cos \left(\frac{1}{42} \left(t + T\right)\right)$

$= \sin \left(\frac{t}{30} + \frac{T}{30}\right) + \cos \left(\frac{t}{42} + \frac{T}{42}\right)$

$= \sin \left(\frac{t}{30}\right) \cos \left(\frac{T}{30}\right) + \cos \left(\frac{t}{30}\right) \sin \left(\frac{T}{30}\right) + \cos \left(\frac{t}{42}\right) \cos \left(\frac{T}{42}\right) - \sin \left(\frac{t}{42}\right) \sin \left(\frac{T}{42}\right)$

Comparing,

$f \left(t\right) = f \left(t + T\right)$

$\left\{\begin{matrix}\cos \left(\frac{T}{30}\right) = 1 \\ \sin \left(\frac{T}{30}\right) = 0 \\ \cos \left(\frac{T}{42}\right) = 1 \\ \sin \left(\frac{T}{42}\right) = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{T}{30} = 2 \pi \\ \frac{T}{42} = 2 \pi\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}T = 60 \pi \\ T = 84 \pi\end{matrix}\right.$

The LCM of $60 \pi$ and $84 \pi$ is

$= 420 \pi$

The period is $T = 420 \pi$

graph{sin(x/30)+cos(x/42) [-83.8, 183.2, -67.6, 65.9]}