Question

Why are the two graph `f(x)=sin(arctan2x)` and `g(x)=(2x)/(sqrt(1+4x^2))` equal?

Answer 1

Please see below

Explanation:

`arctan2x` stands for an angle say `A`, whose tangent ratio is `2x`. In other words, `arctan2x=A` means `tanA=2x`.

As `tanA=2x`, `sin(arctan2x)=sinA`

= `sqrt(1-cos^2A)`

= `sqrt(1-1/sec^2A)`

= `sqrt(1-1/(1+tan^2A))`, but `tanA=2x`, hence

`sin(arctan2x)=sqrt(1-1/(1+(2x)^2))`

or `sin(arctan2x)=sqrt(1-1/(1+4x^2))`

or `sin(arctan2x)=sqrt((4x^2)/(1+4x^2))`

or `sin(arctan2x)=(2x)/sqrt(1+4x^2)`

Hence the graph of `f(x)=sin(arctan2x)` and `g(x)=(2x)/sqrt(1+4x^2)`

are same and appears as follows.
graph{(2x)/sqrt(1+4x^2) [-10, 10, -5, 5]}