# What is the period of f(t)=sin( 4 t )+ cos( (7t)/24 ) ?

Jul 17, 2016

$48 \pi$

#### Explanation:

The period for sin kt and cos kt = (2 pi)/k.

Here, the separate periods for $\sin 4 t \mathmr{and} \cos \left(\frac{7 t}{24}\right)$ are

${P}_{1} = \left(\frac{1}{2}\right) \pi \mathmr{and} {P}_{2} = \left(\frac{7}{12}\right) \pi$

For the compounded oscillation

$f . \left(t\right) = \sin 4 t + \cos \left(\frac{7 t}{24}\right)$,

If t is increased by the least possible period P ,

f(t+P) =f(t).

Here, ( the least possible ) P= 48 pi= (2 X 48)P_1=((12/7) X 48) P2.

$f \left(t + 48 \pi\right) = \sin \left(4 \left(t + 48 \pi\right)\right) + \cos \left(\left(\frac{7}{24}\right) \left(t + 48 \pi\right)\right)$

$= \sin \left(4 t + 192 \pi\right) + \cos \left(\left(\frac{7}{24}\right) t + 14 \pi\right)$

$= \sin 4 t + \cos \left(\frac{7}{12}\right) t$

$= f \left(t\right)$

Note that $14 \pi$ is the least possible multiple of (2pi)#.