The period of the sum of 22 periodic functions is the LCM of their periods.
The period of sin(2x)sin(2x) is T_1=2/2pi=piT1=22π=π
The period of cos(4x)cos(4x) is T_2=2/4pi=1/2piT2=24π=12π
The LCM of T_1T1 and T_2T2 is T=piT=π
To calculate the amplitude, we need the maximum and the minimum of the funtion yy
y=sin2x+cos4xy=sin2x+cos4x
dy/dx=2cos2x-4sin4xdydx=2cos2x−4sin4x
=2cos2x-4*2sin2xcos2x=2cos2x−4⋅2sin2xcos2x
=2cos2x(1-4sin2x)=2cos2x(1−4sin2x)
The max. and min. when dy/dx=0dydx=0
That is,
2cos2x(1-4sin2x)=02cos2x(1−4sin2x)=0
=>⇒
cos2x=0cos2x=0, =>⇒, 2x=pi/22x=π2 or 2x=3/2pi2x=32π
=>⇒, x=pi/4x=π4 or x=3/4pix=34π
and
1-4sin2x=01−4sin2x=0, =>⇒, sin2x=1/4sin2x=14
=>⇒, 2x=0.2532x=0.253, =>⇒, x=0.126x=0.126
So,
y(pi/4)=sin(2*pi/4)+cos(4*pi/4)=1-1=0y(π4)=sin(2⋅π4)+cos(4⋅π4)=1−1=0
y(3/4pi)=sin(2*3/4pi)+cos(4*3/4pi)=-1-1=-2y(34π)=sin(2⋅34π)+cos(4⋅34π)=−1−1=−2
y(0.126)=sin(2*0.126)+cos(4*0.126)=1.125y(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]}
