Question

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

Answer 1

Period `P = pi/3` and the frequency `1/P = 3/pi=0.955`, nearly.
See the oscillation in the graph, for tthe compounded wave, within one period `t in [-pi/6, pi/6]`.

Explanation:

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

Here, the separate periods of the two terms are

`P_1=pi/9 and P_2=pi/21`, respectively..

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

given by

`f(t)=f (t+P) = sin (18(t+LP_1))-cos(42(t+MP_2))`,

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

`LP_1=MP_2=L/9pi=M/21pi=P`.

For`L = 3 and M = 7, P =pi/3`.

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

See how it works.

`f(t+pi/3)=sin(18(t+pi/3))-cos(21(t+pi/3))=sin(18t+6pi)-cos(21t+14pi)`

`=f(t).`

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

`f(t+P/2)=sin(16t+3pi)-cos(21t+7pi)=-sin 18t-+cos 21t ne f(t)`

The frequency`= 1/P=3/pi`.