# What is the frequency of f(theta)= sin 5 t - cos 35 t ?

Aug 7, 2018

$\frac{2}{5} \pi$

#### Explanation:

$f \left(t\right) = \sin 5 t - \cos 35 t$. Let

${p}_{1}$ = period of $\sin 5 t = \frac{2 \pi}{5} \mathmr{and}$

${p}_{2}$ = period of $- \cos 35 t = \frac{2 \pi}{35}$

Now,

the period ( least possible ) P of $f \left(t\right)$ has to be satisfy

$P = {p}_{1} L + {p}_{2} M$

$= \frac{2}{5} L \pi = \frac{2}{35} M$ such tjat

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

As 5 is a factor of 35, their LCM = 35 and

$35 P = 14 L \pi = 2 M \pi \Rightarrow L = 1 , M = 7 \mathmr{and} P = \frac{14}{35} \pi = \frac{2}{5} \pi$

See that $f \left(t + \frac{2}{5} \pi\right) = \sin \left(5 t + 2 \pi\right) - \cos \left(35 t + 14 \pi\right)$

$= \sin 4 t - \cos 35 t = f \left(t\right)$ and that

$f \left(t + \frac{P}{2}\right) = \sin \left(5 t + \pi\right) - \cos \left(35 t + 7 \pi\right)$

$= - \sin 5 t + \cos 35 t$

$\ne f \left(t\right)$
See graph.
graph{(y- sin (5x) + cos (35x))(x-pi/5 +.0001y)(x+pi/5 +0.0001y)=0[-1.6 1.6 -2 2]}

Observe the lines $x = \pm \frac{\pi}{5} = \pm 0.63$, nearly, to mark the period.

For better visual effect, the graph is not on uniform scale.