# What is the period of f(t)=sin( t /2 )+ cos( (7t)/24 ) ?

Apr 21, 2018

The period is $= 48 \pi$

#### Explanation:

A periodic function is such that

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

where $T$ is the period

Therefore,

$\sin \left(\frac{7}{24} t\right) = \sin \left(\frac{7}{24} \left(t + T\right)\right)$

$= \sin \left(\frac{7}{24} t + \frac{7}{24} T\right)$

$= \sin \left(\frac{7}{24} t\right) \cos \left(\frac{7}{24} T\right) + \sin \left(\frac{7}{24} T\right) \cos \left(\frac{7}{24} t\right)$

Comparing the $L H S$ and the $R H S$

$\left\{\begin{matrix}\cos \left(\frac{7}{24} T\right) = 1 \\ \sin \left(\frac{7}{24} T\right) \cos \left(\frac{7}{24} t\right) = 0\end{matrix}\right.$

$\iff$, $\left\{\left(\frac{7}{24} T = 14 \pi\right)\right.$

$T = 48 \pi$

$\sin \left(\frac{t}{2}\right) = \sin \left(\frac{1}{2} \left(t + T\right)\right)$

$= \sin \left(\frac{1}{2} t + \frac{1}{2} T\right)$

$= \sin \left(\frac{t}{2}\right) \cos \left(\frac{T}{2}\right) + \sin \left(\frac{T}{2}\right) \cos \left(\frac{t}{2}\right)$

Comparing the $L H S$ and the $R H S$

$\left\{\begin{matrix}\cos \left(\frac{T}{2}\right) = 1 \\ \sin \left(\frac{T}{2}\right) \cos \left(\frac{t}{2}\right) = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{T}{2} = 2 \pi \\ \frac{T}{2} = 0\end{matrix}\right.$

$T = 4 \pi$

The $L C M$ of $4 \pi \text{and} 48 \pi$ is $= 48 \pi$