An Approach to Mathematic Functions Basics

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

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Section XLVII - Using the Ratio of Areas to define an Angle

Summary for the Section :

• In this Section , we shall look at a slightly different definition of an [ Angle ] ,
  • using the Ratio of Areas instead to define an Angle .

  Thereafter , we shall also work out the Arctangent Function according to this new definition .

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The 360-degrees Symmetry of the Circle :

• Let us first bring in a Circle of radius R ,
  • as the diagram on-the-left below .

  

  Let us now cut-out a [ pie area ] ,

  • as per the diagram on-the-right above .

  And because the Circle does exhibit [ 360-degrees all-round symmetry ] ,

  • we have :
    • 

    i.e. :

    • 

  Consequently :

  • 

    yielding :

  • 

• Let us bring in again the definition of the angle [ Angle Theta ] :
  • 

  we then have :

  • 

• Let us now work out next the Arctangent Function according to this definition of an Angle .

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Working out the Arctangent Function :

• Let us now bring in this set of 3 diagrams below :

  

  The radius of the circle here is then :

  • 

  And the equation expressing [ x ] in terms of [ y ] for any point on the circle is then :

  • 

• Let us now focus for the moment on the diagram on-the-right above .

  We see here that :

  • 

  And the area for [ Area ADCE ] is given by :

  • 

    yielding :

  • 

  The area of [ Triangle ACE ] is then :

  • 

• We then have :
  • 

  We can then plug this into the original definition of [ Theta ] , i.e. :

  • 

    to arrive at a numeric value for [ Theta ] .

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go to the next section : Section XLVIII - Some Graphics related to the Arctangent Function

go to the last section : Section XLVI - Deriving the Arctangent Function

return to the MATH Functions Basics HomePage

Original dated 2014-12-07