# What is the period of f(t)=sin( t / 18 )+ cos( (t)/21 ) ?

Aug 11, 2016

$252 \pi$

#### Explanation:

The periods dor both sin kt and cos kt is $2 \frac{\pi}{k}$

Here, the periods of the separate oscillations given by

$\sin \left(\frac{t}{18}\right) \mathmr{and} \cos \left(\frac{t}{21}\right)$ are $36 \pi \mathmr{and} 42 \pi$, respectively,

For the compounded oscillation f(t), the period is given by

the period$P = 36 L \pi = 42 M \pi$, for the least pair of positive

integers L and M. So, $P = 252 \pi$, against L = 7 and M = 6.

See how it works.

$f \left(t + 252 \pi\right)$

$= \sin \left(\frac{t}{18} + 14 \pi\right) + \cos \left(\frac{t}{21} + 12 \pi\right)$

$= \sin \left(\frac{t}{18}\right) + \cos \left(\frac{t}{21}\right)$

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

Note that when this P is halved, the first term would change its sign..

Aug 11, 2016

$252 \pi$

#### Explanation:

Period of $\sin \left(\frac{t}{18}\right)$ --> $18 \left(2 \pi\right) = 36 \pi$
Period of $\cos \left(\frac{t}{21}\right)$ --> $21 \left(2 \pi\right) = 42 \pi$
Least common multiple of $36 \pi \mathmr{and} 42 \pi$
$\left(36 \pi\right)$ ... x (7) ---> $252 \pi$
$\left(42 \pi\right)$...x (6) ---> $252 \pi$
Period of f(t) --> $252 \pi$