What is the frequency of #f(theta)= sin 24 t - cos 42 t #?

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Answer 1 · 2018-07-28T13:46:35

The frequency is #f=3/pi#

The period #T# of a periodic function #f(x)# is given by

#f(x)=f(x+T)#

Here,

#f(t)=sin24t-cos42t#

Therefore,

#f(t+T)=sin24(t+T)-cos42(t+T)#

#=sin(24t+24T)-cos(42t+42T)#

#=sin24tcos24T+cos24tsin24T-cos42tcos42T+sin42tsin42T#

Comparing,

#f(t)=f(t+T)#

#{(cos24T=1),(sin24T=0),(cos42T=1),(sin42T=0):}#

#<=>#, #{(24T=2pi),(42T=2pi):}#

#<=>#, #{(T=1/12pi=7/84pi),(T=4/84pi):}#

The LCM of #7/84pi# and #4/84pi# is

#=28/84pi=1/3pi#

The period is #T=1/3pi#

The frequency is

#f=1/T=1/(1/3pi)=3/pi#

graph{sin(24x)-cos(42x) [-1.218, 2.199, -0.82, 0.889]}