# What is the period of f(t)=sin( (t)/4 )+ cos( (t)/12 ) ?

Apr 17, 2016

$24 \pi$

#### Explanation:

The period of both sin kt and cos kt is $\frac{2 \pi}{k}$.
For the separate oscillations given by $\sin \left(\frac{t}{4}\right) \mathmr{and} \cos \left(\frac{t}{12}\right)$, the periods are $8 \pi \mathmr{and} 24 \pi$, respectively.

So. for the compounded oscillation given by $\sin \left(\frac{t}{4}\right) + \cos \left(\frac{t}{12}\right)$, the period is the LCM = $24 \pi$.

In general, if the separate periods are ${P}_{1} \mathmr{and} {P}_{2}$, the period for the compounded oscillation is from $m {P}_{1} = n {P}_{2}$, for the least positive-integer pair [m, n].

Here, ${P}_{1} = 8 \pi \mathmr{and} {P}_{2} = 24 \pi$. So, m = 3 and n = 1.