What is the period of #f(t)=sin( t /2 )+ cos( (7t)/24 ) #?

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Answer 1 · 2018-04-21T06:41:33

The period is #=48pi#

A periodic function is such that

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

where #T# is the period

Therefore,

#sin(7/24t)=sin(7/24(t+T))#

#=sin(7/24t+7/24T)#

#=sin(7/24t)cos(7/24T)+sin(7/24T)cos(7/24t)#

Comparing the #LHS# and the #RHS#

#{(cos(7/24T)=1),(sin(7/24T)cos(7/24t)=0):}#

#<=>#, #{(7/24T=14pi):}#

#T=48pi#

#sin(t/2)=sin(1/2(t+T))#

#=sin(1/2t+1/2T)#

#=sin(t/2)cos(T/2)+sin(T/2)cos(t/2)#

Comparing the #LHS# and the #RHS#

#{(cos(T/2)=1),(sin(T/2)cos(t/2)=0):}#

#<=>#, #{(T/2=2pi),(T/2=0):}#

#T=4pi#

The #LCM# of #4pi "and" 48pi# is #=48pi#