On the 30-Rhombus Symmetric Solid Object

by Frank C. Fung ( 1st published in February, 2013. )

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Section XVIII - Bringing-in the Tangent Addition Formula

Summary for the Section :

• In this and the next 2 Sections , we shall attempt to establish :
  • an expression for the [ Tangent of 9-degrees ] based of [ radicals ] .

  And in this Section , we shall bring-in the Tangent Addition Formula , i.e. :

  • 

  and :

  • develop further formulas for the tangents of [ integer-multiples of any fixed-angle Theta ] ,
    • for our use in the next 2 Sections .

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Bringing-in the Tangent Addition Formula :

• Let us first bring-in the Tangent Addition Formula , in its usual format , as follows :
  • 

  We can then set up :

  • the [ Angle Theta ] as any angle of a specific value ,

    and

  • the value [ T ] as the value for the [ Tangent of Theta ] .

  We then have :

  • 

• Consequently , based on the Tangent Addition Formula above , we have :
  • 

  • 

  • 

  • 

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Bringing-in the Pascal Triangle :

• Let us now take a quick look at the last equation above , i.e. this equation :
  • 

  First :

  • we notice here that the co-efficients in the [ nominator of the Fraction ] are :
    • [ +5 ] / [ -10 ] / [ +1 ] respectively ;

    and :

    • these are co-efficients on the [ 6th row ] of the Pascal Triangle ,
      • as marked-off in [ cyan / light-green colors ] in the diagram on-the-right below ;

      with the [ signs ] of the co-efficients here alternating .

  

  Secondly :

  • we also notice here that the co-efficients in the [ denominator of the Fraction ] are :
    • [ +1 ] / [ -10 ] / [ +5 ] respectively ;

    and :

    • these are also co-efficients on the [ 6th row ] of the Pascal Triangle ,
      • as marked-off in [ red / magenta colors ] in the diagram on-the-right below ;

      with the [ signs ] of the co-efficients here alternating .

  

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Setting up the { 4-color Pascal Triangle }:

• Let us now bring-in the { 4-color Pascal Triangle } ,
  • as per the diagram on-the-right below .

  

  Let us also bring-in , at this point , the formula for the [ Tangent of 9-Theta ] , i.e. :

  • 

• We now make this observation :
  • the co-efficients in the nominator of the Fraction above are :
    • [ +9 ] / [ -84 ] / [ +126 ] / [ -36 ] / [ +1 ] respectively ,

    and :

    • these are as marked-off in [ cyan / light-green colors ] on the [ 10th row ] in the diagram on-the-right above ;

  • the co-efficients in the denominator of the Fraction above are :
    • [ +1 ] / [ -36 ] / [ +126 ] / [ -84 ] / [ +1 ] respectively ,

    and :

    • these are as marked-off in [ red / magenta colors ] on the [ 10th row ] in the diagram on-the-right above .

  Our contention here is that :

  • the above [ scheme-of-things ] on the Nominator / Denominator for the { Tangent Formula for [ N*Theta ] }
    • will remain valid throughout the Pascal Triangle .

• But this is not the subject matter of this paper , and :
  • we might come back for a full discussion of this observation later in another paper .

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go to the next section : Section XIX - Setting up the Equation for the [ Tangent of 9-degrees ]

go to the last section : Section XVII - Reason for calculating the [ Tangent of 9-degrees ]

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04