What is the frequency of f(theta)= sin 24 t - cos 42 t ?

Jul 28, 2018

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

Explanation:

The period $T$ of a periodic function $f \left(x\right)$ is given by

$f \left(x\right) = f \left(x + T\right)$

Here,

$f \left(t\right) = \sin 24 t - \cos 42 t$

Therefore,

$f \left(t + T\right) = \sin 24 \left(t + T\right) - \cos 42 \left(t + T\right)$

$= \sin \left(24 t + 24 T\right) - \cos \left(42 t + 42 T\right)$

$= \sin 24 t \cos 24 T + \cos 24 t \sin 24 T - \cos 42 t \cos 42 T + \sin 42 t \sin 42 T$

Comparing,

$f \left(t\right) = f \left(t + T\right)$

$\left\{\begin{matrix}\cos 24 T = 1 \\ \sin 24 T = 0 \\ \cos 42 T = 1 \\ \sin 42 T = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}24 T = 2 \pi \\ 42 T = 2 \pi\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}T = \frac{1}{12} \pi = \frac{7}{84} \pi \\ T = \frac{4}{84} \pi\end{matrix}\right.$

The LCM of $\frac{7}{84} \pi$ and $\frac{4}{84} \pi$ is

$= \frac{28}{84} \pi = \frac{1}{3} \pi$

The period is $T = \frac{1}{3} \pi$

The frequency is

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

graph{sin(24x)-cos(42x) [-1.218, 2.199, -0.82, 0.889]}