# What is the frequency of f(theta)= sin 24 t - cos 45 t ?

Aug 31, 2016

$\frac{1}{30 \pi}$

#### Explanation:

Frequency=1/(period)

The epriod for both sin k t and cos kt is $\frac{2}{k} \pi$.

So, the separate periods for the oscillations sin 24t and cos 45t are

$\frac{2}{12} \pi \mathmr{and} \frac{2}{45} \pi$.

The period P for the compounded oscillation

$f \left(t\right) = \sin 24 t - \cos 45 t$ is given by

$P = M \left(\frac{2}{24} \pi\right) = N \left(\frac{2}{45} \pi\right)$, where M and N make P the least

positive integer multiple of $2 \pi$.

Easily, M= 720 and N=675, making P = 30pi.

So, the frequency $\frac{1}{P} = \frac{1}{30 \pi}$.

See how P is least.

$f \left(t + P\right)$

$= f \left(t + 30 \pi\right)$

=sin (24(t+30pi)-cos(45(t+30pi)

$= \sin \left(24 t + 720 \pi\right) - \cos \left(45 t + 1350 i\right)$

$= \sin 24 t - \cos 45 t$

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

Here, if Pis halved to $15 \pi$, the second term would become

$-$cos (45t+odd multiple of pi)#

$= + \cos 45 t$