What is the period of #f(t)=sin( t / 30 )+ cos( (t)/ 42) #?

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Answer 1 · 2018-08-01T09:42:21

The period is #T=420pi#

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

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

Here,

#f(t)=sin(t/30)+cos(t/42)#

Therefore,

#f(t+T)=sin(1/30(t+T))+cos(1/42(t+T))#

#=sin(t/30+T/30)+cos(t/42+T/42)#

#=sin(t/30)cos(T/30)+cos(t/30)sin(T/30)+cos(t/42)cos(T/42)-sin(t/42)sin(T/42)#

Comparing,

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

#{(cos(T/30)=1),(sin(T/30)=0),(cos(T/42)=1),(sin(T/42)=0):}#

#<=>#, #{(T/30=2pi),(T/42=2pi):}#

#<=>#, #{(T=60pi),(T=84pi):}#

The LCM of #60pi# and #84pi# is

#=420pi#

The period is #T=420pi#

graph{sin(x/30)+cos(x/42) [-83.8, 183.2, -67.6, 65.9]}