# What is the period of f(t)=sin( t / 30 )+ cos( (t)/ 33) ?

Jul 28, 2016

$660 \pi$

#### Explanation:

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

So, the separate periods for the the two terms in f(t) are

$60 \pi \mathmr{and} 66 \pi$

The period for the compounded oscillation of f(t) is given by

least positive integer multiples L and M such that

the period P = 60 L = 66 M.

L = 11 and M =10 for P =660$\pi$.

See how it works.

$f \left(t + P\right)$

$= f \left(t + 660 \pi\right)$

$= \sin \left(\frac{t}{30} + 22 \pi\right) + \cos \left(\frac{t}{33} + 20 \pi\right)$

$= \sin \left(\frac{t}{30}\right) + \cos \left(\frac{t}{33}\right)$

$= f \left(t\right)$.

Note that, $\frac{P}{2} = 330 \pi$ is not a period, for the sine term.