# What is the period of f(t)=sin( t / 30 )+ cos( (t)/ 12 ) ?

Jul 28, 2016

$120 \pi$

#### Explanation:

The period for both $\sin k \pi \mathmr{and} \cos k \pi i s$(2pi)/k.

Here, the separate periods for terms in f(t) are $60 \pi \mathmr{and} 24 \pi$

So, the period P for the compounded oscillation is given by

P = 60 L = 24 M, where L and M together form the least possible pair

of positive integers. L= 2 and M = 10 and the compounded period

$P = 120 \pi$.

See how it works.

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

$= f \left(t + 120 \pi\right)$

$= \sin \left(\frac{t}{30} + 4 \pi\right) + \cos \left(\frac{t}{12} + 10 \pi\right)$

$= \sin \left(\frac{t}{30}\right) + \cos \left(\frac{t}{12}\right)$

=f(t).

Note that $\frac{P}{20} = 50 \pi$ is not a period, for the cosine term.