An Approach to Mathematic Functions Basics

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

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Section XLIII - Tangent Additions and the Pascal Triangle

Summary for the Section :

• In this Section , we shall attempt to show that :
  • for the [ fraction ] expressing the { tangent of [ N*Theta ] } in terms of the { tangent of [ Theta ] } ,
    • the co-efficients in the Nominator and the co-efficients in the Denominator
      • are always members of the same row of the Pascal Triangle .

  And we shall also create the { 4-Colors Pascal Triangle } , as per the diagram on-the-right below , for our use in the next Section .

  

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Doing a Quick Example with the { Tangent of [ 8*Theta ] } :

• Let us now quickly bring-in the formula for the { Tangent of [ 8*Theta ] } :
  • 

    where :

    • 

  And we bring-in also this set of 2 diagrams below for a quick comparison .

  

  And we see here that :

  • FIRST :
    • The co-efficients in the Nominator , namely [ +8 / -56 / +56 / -8 ] ,
      • are members of [ Row 9 ] of the Pascal Triangle ,
        • as marked-off in [ cyan-color ] and [ green-color ] in the diagram on-the-left above .

  • SECOND :
    • The co-efficients in the Denominator , namely [ +1 / -28 / +70 / -28 / +1 ] ,
      • are members of [ Row 9 ] of the Pascal Triangle ,
        • as marked-off in [ red-color ] and [ magenta color ] in the diagram on-the-right above .

• Let us now make a general statement on the pattern of these co-efficients , next .

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Re-counting the Co-efficients :

• Let us bring-in again the results of our calculation process in the last Section and we have :
  • 

  • 

  • 

  • 

  • 

  • 

  • 

  • 

  And we bring in also this set of 2 diagrams on the Pascal Triangle for our demonstration purposes :

  

  And we see here that :

  • for the [ fraction ] expressing the { Tangent of [ N*Theta ] } in terms of the { Tangent of [ Theta ] } :

    FIRST :

    • The co-efficients in the Nominator and the Denominator are members of [ Row N+1 ] of the Pascal Triangle .

    SECOND :

    • The co-efficients in the Nominator are as marked-off [ cyan-color ] and [ green-color ] in the diagram on-the-left above ,

      with :

      • [ cyna-color ] members spotting [ positive signs ] and [ green-color ] members spotting [ negative signs ] .

    THIRD :

    • The co-efficients in the Denominator are as marked-off [ red-color ] and [ magenta-color ] in the diagram on-the-right above ,

      with :

      • [ red-color ] members spotting [ positive signs ] and [ magenta-color ] members spotting [ negative signs ] .

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Creating the { 4-Colors Pascal Triangle } :

• Let us now consolidate the 2 diagrams above into single diagram ,
  • to create the { 4-Colors Pascal Triangle } ,
    • as the diagram on-the-right below .

  

• This is an interesting diagram which we shall put into good use in the next Section .

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go to the next section : Section XLIV - Algebraic Equations arising from { Tangent of [ Pi-over-4 ] = 1 }

go to the last section : Section XLII - The Tangent Addition Formula in Action

return to the MATH Functions Basics HomePage

Original dated 2014-12-07