# What is the frequency of f(theta)= sin 18 t - cos 42 t ?

Jan 9, 2017

Period $P = \frac{\pi}{3}$ and the frequency $\frac{1}{P} = \frac{3}{\pi} = 0.955$, nearly.
See the oscillation in the graph, for tthe compounded wave, within one period $t \in \left[- \frac{\pi}{6} , \frac{\pi}{6}\right]$.

#### Explanation:

graph{sin (18x)-cos (12x) [-0.525, 0.525 -2.5, 2.5]} The period of both sin kt and cos kt is $\frac{2}{k} \pi$.

Here, the separate periods of the two terms are

${P}_{1} = \frac{\pi}{9} \mathmr{and} {P}_{2} = \frac{\pi}{21}$, respectively..

The period ( least possible ) P, for the compounded oscillation, is

given by

$f \left(t\right) = f \left(t + P\right) = \sin \left(18 \left(t + L {P}_{1}\right)\right) - \cos \left(42 \left(t + M {P}_{2}\right)\right)$,

for least possible ( positive ) integer multiples L and M such that

$L {P}_{1} = M {P}_{2} = \frac{L}{9} \pi = \frac{M}{21} \pi = P$.

For$L = 3 \mathmr{and} M = 7 , P = \frac{\pi}{3}$.

Note that P/2 is not the period, so that P is the least possible value.

See how it works.

$f \left(t + \frac{\pi}{3}\right) = \sin \left(18 \left(t + \frac{\pi}{3}\right)\right) - \cos \left(21 \left(t + \frac{\pi}{3}\right)\right) = \sin \left(18 t + 6 \pi\right) - \cos \left(21 t + 14 \pi\right)$

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

Check by back substiution P/2, instead of P, for least P.

$f \left(t + \frac{P}{2}\right) = \sin \left(16 t + 3 \pi\right) - \cos \left(21 t + 7 \pi\right) = - \sin 18 t - + \cos 21 t \ne f \left(t\right)$

The frequency$= \frac{1}{P} = \frac{3}{\pi}$.