Deriving the Arctangent Function

An Approach to Mathematic Functions Basics

Section XLVI - Deriving the Arctangent Function

In this Section , we shall go thru the derivation process for the Arctangent Function :

It is appropriate , at this first instance , to do a quick review of the the usual definition of an [ Angle ] .

And we bring-in this set of 3 diagrams below for our quick reference :

In general , [ Angle Theta ] is defined by :
- the ratio of the [ Arc-Length sustained by the Angle ] to the [ radius of the Circle ] ,
  - as per the diagram in-the-middle above .
And the [ Tangent of Theta ] is defined via the triangle { Triangle ABC }
- as shown in the diagram on-the-left above .
The diagram on-the-right above then shows the physical relation between :
- [ radius R ] , [ H ] , [ W ] and [ Arc-Length D-C ] .

For the derivation of the Arctangent Function , we shall start-off with this rather simple [ Trigonometric Identity ] :
Taking derivatives with respect to [ Theta ] on both sides then yields us this relation :
- yielding :
Consequently :
- yielding :
And based on this , we can now write :
Let us now set up the variable [ T ] such that :
And we take special note here that :
And with this in-hand , we are now ready to do the integration on the equation immediately above .
Let us now bring-in again the said-equation from above :
Intergrating on both sides then yields us this relation :
- yielding :
And the final equation for the Arctangent Function is then :