# What is the frequency of f(theta)= sin 12 t - cos 33 t ?

Jul 25, 2016

$\frac{1}{22 \pi}$

#### Explanation:

The least positive P for which f(t+P)=f(t) is the period of f(theta)#

Separately, the period of both cos kt and sin kt = $\frac{2 \pi}{k}$.
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Here, the separate periods for periods for sin (12t) and cos (33t) are

$\frac{2 \pi}{12} \mathmr{and} \frac{2 \pi}{33}$.

So, the compounded period is given by $P = L \left(\frac{\pi}{6}\right) = M \left(2 \frac{\pi}{33}\right)$

such that P is positive and least.

Easily, $P = 22 \pi$, for L= 132 and M = 363.

The frequency $= \frac{1}{P} = \frac{1}{22 \pi}$

You can see how this works.

$f \left(t + 22 \pi\right)$

$= \sin \left(12 \left(t + 22 \pi\right)\right) - \cos \left(33 \left(t + 22 \pi\right)\right)$

$= \sin \left(12 t + 264 \pi\right) - \cos \left(33 t + 866 \pi\right)$

$= \sin 12 t - \cos 33 t$

$= f \left(t\right)$

You can verify that $\frac{P}{2} = 11 \pi$ is not a period., for the cosine term in

f(t). P has to be a period for every term in such compounded

oscillations.