What is the frequency of #f(theta)= sin 24 t - cos 45 t #?

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Answer 1 · 2016-08-31T05:23:35

#1/(30pi)#

Frequency=1/(period)#

The epriod for both sin k t and cos kt is #2/kpi#.

So, the separate periods for the oscillations #sin 24t and cos 45t are

#2/12pi and 2/45pi#.

The period P for the compounded oscillation

#f(t)=sin 24t-cos 45t # is given by

#P = M(2/24pi)=N(2/45pi)#, where M and N make P the least

positive integer multiple of #2pi#.

Easily, M= 720 and N=675, making P = 30pi#.

So, the frequency #1/P = 1/(30pi)#.

See how P is least.

#f(t+P)#

#=f(t+30pi)#

#=sin (24(t+30pi)-cos(45(t+30pi)#

#= sin (24t+720pi)-cos(45t+1350i)#

#=sin 24t-cos45t#

#=f(t)#.

Here, if Pis halved to #15pi#, the second term would become

#-#cos (45t+odd multiple of #pi)#

#=+cos 45t#