On a Quick Expansion Method for Pythagorus Triples

by Frank C. Fung ( 1st published in December, 2021. )

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Section XXXIII - Developing the One-On-One Expansion Formula ( III )

Summary for the Section :

• In this Section , we shall re-express [ P ] and [ Q ] in terms of 3 new { variables / fixed values } , namely :
  • [ OMEGA ] , [ LAMBDA ] and [ THETA ] ;

    for the purpose of facilitating the calculation of [ P-Square ] and [ Q-Square ] in the next Section .

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Setting up [ OMEGA ] / [ LAMBDA ] / [ THETA ] :

• For our analysis to follow here and in the next 2 Sections ,
  • we shall now set up 3 new { variables / fixed-values } , namely :
    • [ OMEGA ] , [ LAMBDA ] and [ THETA ] .

  FIRST , we set up [ OMEGA ] :

  • We simply have :
    • 

  SECOND , we set up [ LAMBDA ] and [ THETA ] :

  • We take note here , first-of-all :
    • 

    Therefore ,

    • 

    Consequently , we have this relation :

    • 

    And we take note here that the above equation depicts a Pythagrus Triple relation of the format :

    • [ C-Square ] = [ A-Square ] + [ B-Square ]

      and therefore involves a { Right-Angle Triangle } .

  • As such , we can now set up [ LAMBDA ] :
    • 

      being the length of the Hypotunuse ,

    and also the angle [ Angle THETA ] given by :

    • 

    Thus :

    • the width of the { Right-Angle Triangle } is given by :
      • 

    • and the height of the { Right-Angle Triangle } is given by :
      • 

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Converting [ P ] :

• We then bring-in from the last Section - Section XXXII , this expression for [ P ] :
  • 
    

  And on expanding terms , we have :

  • 
    

• Let us now bring in this set of relations from above for the conversion :
  • 

  • 

  • 

  • 

  And on applying this to the expression for [ P ] immediately above , we have :

  • 
    

• And the above expression for [ P ] is then the expression we shall be using for calculating [ P-Square ] in the next Section .

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Converting [ Q ] :

• We then bring-in from the last Section - Section XXXII , this expression for [ Q ] :
  • 
    

  And on expanding terms , we have :

  • 
    

• Let us now bring in this relation from above :
  • 

    in the format of { [ C-Square ] = [ A-Square ] + [ B-Square ] } .

  Consequently , we can now express the very last term in the 2nd row above in this manner :

  • 

  And on applying this to the expression for [ Q ] immediately above , we then have :

  • 
    

  And on consolidating terms , we have :

  • 
    

• Let us now bring in this set of relations from above for the conversion :
  • 

  • 

  • 

  • 

  And on applying this to the expression for [ Q ] immediately above , we have :

  • 
    

• And the above expression for [ Q ] is then the expression we shall be using for calculating [ Q-Square ] in the next Section .

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What's Next :

• And with this in-hand , we shall take a closer look at { [ P-Square ] + [ Q-Square ] } , next .

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go to the next Section : Section XXXIV - Developing the One-On-One Expansion Formula ( IV )

go to the last Section : Section XXXII - Developing the One-On-One Expansion Formula ( II )

return to the Pythagorus Triples Homepage

Original dated 2021-12-27