Question

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

Answer 1

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

Explanation:

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]}