arctan2xarctan2x stands for an angle say AA, whose tangent ratio is 2x2x. In other words, arctan2x=Aarctan2x=A means tanA=2xtanA=2x.
As tanA=2xtanA=2x, sin(arctan2x)=sinAsin(arctan2x)=sinA
= sqrt(1-cos^2A)√1−cos2A
= sqrt(1-1/sec^2A)√1−1sec2A
= sqrt(1-1/(1+tan^2A))√1−11+tan2A, but tanA=2xtanA=2x, hence
sin(arctan2x)=sqrt(1-1/(1+(2x)^2))sin(arctan2x)=√1−11+(2x)2
or sin(arctan2x)=sqrt(1-1/(1+4x^2))sin(arctan2x)=√1−11+4x2
or sin(arctan2x)=sqrt((4x^2)/(1+4x^2))sin(arctan2x)=√4x21+4x2
or sin(arctan2x)=(2x)/sqrt(1+4x^2)sin(arctan2x)=2x√1+4x2
Hence the graph of f(x)=sin(arctan2x)f(x)=sin(arctan2x) and g(x)=(2x)/sqrt(1+4x^2)g(x)=2x√1+4x2
are same and appears as follows.
graph{(2x)/sqrt(1+4x^2) [-10, 10, -5, 5]}
