An Approach to Mathematic Functions Basics

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

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Section XLII - The Tangent Addition Formula in Action

Summary for the Section :

• In this Section , we shall attempt to express :
  • the { tangent of [ N * Theta ] } in terms of the { tangent of [ Theta ] } .

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Setting-up the value [ T ] as the { tangent of [ Theta ] } :

• Let us first set up [ T ] as the value of the { tangent of [ Theta ] } , i.e. :
  • 

  And we bring-in again the Tangent Addition Formula :

  • 

    for our applications below .

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The Tangent Addition Formula in action :

• First-of-all , for the { tangent of [ 2*Theta ] } :
  • we can simply substitute into the Tangent Addition Formula :
    • { [ Alpha ] = [ Theta ] } and { [ Beta ] = [ Theta ] }

    to arrive at :

    • 

• And :
  • For the { Tangent of [ 3*Theta ] } , we can use :
    • { [ Alpha ] = [ 2*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

  • For the { Tangent of [ 4*Theta ] } , we can use :
    • { [ Alpha ] = [ 3*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

  • For the { Tangent of [ 5*Theta ] } , we can use :
    • { [ Alpha ] = [ 4*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

  • For the { Tangent of [ 6*Theta ] } , we can use :
    • { [ Alpha ] = [ 5*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

  • For the { Tangent of [ 7*Theta ] } , we can use :
    • { [ Alpha ] = [ 6*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

        yielding :

      • 

  • For the { Tangent of [ 8*Theta ] } , we can use :
    • { [ Alpha ] = [ 7*Theta ] } and { [ Beta ] = [ Theta ] }

      to arrive at :

      • 

        yielding :

      • 

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Summarizing :

• Let us now quickly summarize the results of our calculation process above .

  And we have :

  • 

  • 

  • 

  • 

  • 

  • 

  • 

  • 

• QUESTION :
  • Do we see a pattern here ?

  And

  • the ANSWER is yes !

  More details on this in the next Section coming up .

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go to the next section : Section XLIII - Tangent Additions and the Pascal Triangle

go to the last section : Section XLI - Deriving the Tangent Aditiona Formula

return to the MATH Functions Basics HomePage

Original dated 2014-12-07