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

1 Answer
Jan 9, 2017

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#.