# What is the frequency of f(theta)= sin 24 t - cos 15 t ?

Feb 19, 2017

The frequency is $f = \frac{3}{2 \pi}$

#### Explanation:

Let 's calculate the period $T$

The period of the sum of 2 periodic functions is the $L C M$ of their periods.

The period of $\sin 24 t$ is $= \frac{2}{24} \pi = \frac{\pi}{12}$

The period of $\cos 15 t$ is $= \frac{2}{15} \pi$

The multiples of $\frac{\pi}{12}$ are

$\left\{\frac{\pi}{12} , \frac{2}{12} \pi , \frac{3}{12} \pi , \frac{4}{12} \pi , \frac{5}{12} \pi , \frac{6}{12} \pi , \frac{7}{12} \pi , \textcolor{red}{\frac{8}{12} \pi} , \frac{9}{12} \pi \ldots\right\}$

The multiples of $\frac{2}{15} \pi$ are

$\left\{\frac{2}{15} \pi , \frac{4}{15} \pi , \frac{6}{15} \pi , \frac{8}{15} \pi , \textcolor{red}{\frac{10}{15} \pi} , \frac{12}{15} \pi\right\}$

The $L C M$ of $\frac{\pi}{12}$ and $\frac{2}{15} \pi$ is

$\frac{8}{12} \pi = \frac{2}{3} \pi$

So,

$T = \frac{2}{3} \pi$

and the frequency is

$f = \frac{1}{T} = \frac{3}{2 \pi}$