An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in November, 2014. )

Top Of Page

Section XLVIII - Some Graphics related to the Arctangent Function

Summary for the Section :

• In this Section , we shall take a look at some Graphics related to the Arctangent Function .

go to Top Of Page

The Equation within the Integrand :

• Let us bring in again the Arctangent Function :
  • 

  and focus , for the time being , on this equation within the Integrand on the right-hand-side ,

  • but in this format below :
    • 

  And the graph for this equation is then :

  

• And we take note at this point that :
  • 

  Therefore :

  • 

  

go to Top Of Page

Splitting [ Pi-over-4 ] into equal areas :

• Let us now split the [ Pi-over-4 ] area into :
  • 2 equal areas ,
  • 3 equal areas , and
  • 4 equal areas ;

    as per this set of 3 diagrams below .

  

  And we simply take note here that :

  • [ X = 0.414214 ] is the solution to this Algebraic Equation :
    • 

  • [ X = 0.267949 ] is the solution to this Algebraic Equation :
    • 

  • [ X = 0.198912 ] is the solution to this Algebraic Equation :
    • 

• And we also bring in this set of 3 diagrams for whatever-it-may-be-worth :

  

go to Top Of Page

The Lingering Question :

• Our thinking at this point is that :
  • For the values of [ T ] and [ Theta ] in the relation :
    • 

    ONE :

    • If [ T ] is a Fraction less than [ 1 ] ,
      • then the value of [ Theta ] is not expressible as a fraction of [ Pi-over-4 ] .

    TWO :

    • If [ Theta ] is less than [ Pi-over-4 ] and is expressible as a fraction of [ Pi-over-4 ] ,
      • then the value of [ T ] cannot be a fraction .

• TRUE or NOT TRUE ?

go to Top Of Page

go to the next section : Section XLIX - Concluding Remarks

go to the last section : Section XLVII - Using the Ratio of Areas to define an Angle

return to the MATH Functions Basics HomePage

Original dated 2014-12-07