# What is the period of f(t)=sin( t / 14 )+ cos( (t)/21 ) ?

Apr 19, 2016

$84 \pi$

#### Explanation:

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

The periods of the two separate oscillations in f(t) are $28 \pi \mathmr{and} 42 \pi$.

The period for the compounded oscillation is such that 28 M = 42 N, for lowest positive integers M and N. Easily, M =3 and N = 2. These give the LCM common value $84 \pi$ as the period for f(t).
$f \left(t + 84 \pi\right) = \sin \left(\frac{t}{14} + 6 \pi\right) + \cos \left(\frac{t}{21} + 4 \pi\right) = \sin \left(\frac{t}{14}\right) + \cos \left(\frac{t}{21}\right) = f \left(t\right)$