# What is the period of f(t)=sin( t /2 )+ cos( (13t)/24 ) ?

Apr 25, 2016

$52 \pi$

#### Explanation:

The period of both sin kt and cos kt is $\frac{2 \pi}{k}$.

So, separately, the periods of the two terms in f(t) are $4 \pi \mathmr{and} \left(\frac{48}{13}\right) \pi$.

For the sum, the compounded period is given by $L \left(4 \pi\right) = M \left(\left(\frac{48}{13}\right) \pi\right)$, making the common value as the least integer multiple of $\pi$.

L=13 and M=1. The common value = $52 \pi$;

Check: $f \left(t + 52 \pi\right) = \sin \left(\left(\frac{1}{2}\right) \left(t + 52 \pi\right)\right) + \cos \left(\left(\frac{24}{13}\right) \left(t + 52 \pi\right)\right)$
$= \sin \left(26 \pi + \frac{t}{2}\right) + \cos \left(96 \pi + \left(\frac{24}{13}\right) t\right)$
$= \sin \left(\frac{t}{2}\right) + \cos \left(\frac{24}{13} t\right) = f \left(t\right)$..