What is the frequency of #f(theta)= sin 12 t - cos 33 t #?

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Answer 1 · 2016-07-25T06:26:39

#1/(22pi)#

The least positive P for which f(t+P)=f(t) is the period of f(theta)#

Separately, the period of both cos kt and sin kt = #(2pi)/k#.
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Here, the separate periods for periods for sin (12t) and cos (33t) are

#(2pi )/12 and (2pi)/33#.

So, the compounded period is given by #P=L(pi/6)=M(2pi/33) #

such that P is positive and least.

Easily, #P=22pi#, for L= 132 and M = 363.

The frequency #= 1/P =1/(22pi)#

You can see how this works.

#f(t+22pi)#

#=sin(12(t+22pi))-cos(33(t+22pi))#

#=sin(12t+264pi)-cos(33t+866pi)#

#=sin 12t-cos 33t#

#=f(t)#

You can verify that #P/2=11pi# is not a period., for the cosine term in

f(t). P has to be a period for every term in such compounded

oscillations.