# What is the period of f(t)=sin( t / 12 )+ cos( (t)/21 ) ?

May 11, 2016

$168 \pi$.

#### Explanation:

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

Here, the separate periods of oscillation of the waves

$\sin \left(\frac{t}{12}\right) \mathmr{and} \cos \left(\frac{t}{21}\right)$ are $24 \pi \mathmr{and} 42 \pi$.

So, the period for the compounded oscillation for the sun is the

$L C M = 168 \pi$.

You see how it works.

$f \left(t + 168 \pi\right) = \sin \left(\left(\frac{1}{12}\right) \left(t + 168 \pi\right)\right) + \cos \left(\left(\frac{1}{21}\right) \left(t + 168 \pi\right)\right)$

$= \sin \left(\frac{t}{12} + 14 \pi\right) + \cos \left(\frac{t}{21} + 8 \pi\right)$

$= \sin \left(\frac{t}{12}\right) + \cos \left(\frac{t}{21}\right)$

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