How do you find the amplitude and period of a function #y = sin(2x) + cos(4x)#?

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Answer 1 · 2017-06-04T09:27:59

The period of the function is #=pi#
The amplitude is #=1.56#

The period of the sum of #2# periodic functions is the LCM of their periods.

The period of #sin(2x)# is #T_1=2/2pi=pi#

The period of #cos(4x)# is #T_2=2/4pi=1/2pi#

The LCM of #T_1# and #T_2# is #T=pi#

To calculate the amplitude, we need the maximum and the minimum of the funtion #y#

#y=sin2x+cos4x#

#dy/dx=2cos2x-4sin4x#

#=2cos2x-4*2sin2xcos2x#

#=2cos2x(1-4sin2x)#

The max. and min. when #dy/dx=0#

That is,

#2cos2x(1-4sin2x)=0#

#=>#

#cos2x=0#, #=>#, #2x=pi/2# or #2x=3/2pi#

#=>#, #x=pi/4# or #x=3/4pi#

and

#1-4sin2x=0#, #=>#, #sin2x=1/4#

#=>#, #2x=0.253#, #=>#, #x=0.126#

So,

#y(pi/4)=sin(2*pi/4)+cos(4*pi/4)=1-1=0#

#y(3/4pi)=sin(2*3/4pi)+cos(4*3/4pi)=-1-1=-2#

#y(0.126)=sin(2*0.126)+cos(4*0.126)=1.125#

Therefore,

The amplitude is #=(Max-min)/2=(1.125-(-2))/2=1.56#
graph{(y-sin(2x)-cos(4x))=0 [-3.523, 5.245, -2.154, 2.23]}