An Approach to Mathematic Functions Basics

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

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Section XLIV - Algebraic Equations arising from { Tangent of [ Pi-over-4 ] = 1 }

Summary for the Section :

• In this Section , we take note first-of-all that :
  • 

  We shall subsequently set up the angle [ Angle Theta ] such that :

  • 

    where :

    • [ N ] is any { positive integer } greater than or equal to [ 2 ] .

  We shall then set up an algbraic equation to solve for the value of the { tangent of [ Theta ] } ,

  • noting that :
    • 

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Doing a Quick Example with the { Tangent of [ 7*Theta ] } :

• In this quick example , we shall use [ N = 7 ] throughout .

  We then set up :

  • 

  And consequently :

  • 

    and

  • 

• Let us now bring in the formula for the { tangent of [ 7*Theta ] } from the last Section , Section XXXIX :
  • 

    where :

    • 

  Since :

  • 

  we then have :

  • 

  Consequently , we can now write :

  • 

  And the final Algebraic Equation here for [ N = 7 ] is then :

  • 

    where :

    • 

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Counting the Co-efficients for the derived Algebraic Equations :

• It is easy enough to do the derivations for other values of [ N ] and
  • we shall simply list the resultant Algebraic Equations here .

  Then :

  • For [ N = 2 ] , the Algebraic Equation is :
    • 

  • For [ N = 3 ] , the Algebraic Equation is :
    • 

  • For [ N = 4 ] , the Algebraic Equation is :
    • 

  • For [ N = 5 ] , the Algebraic Equation is :
    • 

  • For [ N = 6 ] , the Algebraic Equation is :
    • 

  • For [ N = 7 ] , the Algebraic Equation is :
    • 

  • For [ N = 8 ] , the Algebraic Equation is :
    • 

• QUESTION :
  • Do we see a pattern here ?

  And the ANSWER is YES !

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Bringing in the { 4-Colors Pascal Triangle } :

• Let us now bring in the { 4-Colors Pascal Triangle } from the last Section :
  • as shown on-the-right below .

  

• The general scheme for the co-efficients of each of the Algebraic Equations above is :
  • each Algebraic Equation spots a ROW in the { 4-Colors Pascal Triangle } above ,

    and

  • the color of the members in each ROW determines the sign of the co-efficient :
    • the [ red-color members ] will spot a [ positive sign ] ,
    • the [ cyan-color members ] will spot a [ negative sign ] ,
    • the [ magenta-color members ] will spot a [ negative sign ] , and
    • the [ green-color members ] will spot a [ positive sign ] .

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But where do we go from here :

• Let us now make this rather simple observation :
  • 

  • 

  • 

  • 

    and so-on-and-so-forth .

• Let us now do another example for [ N = 7 ] but using [ 3-Pi-over-4 ] instead .

  We then set up the angle [ Angle Psi ] :

  • 

  And we also set up the value [ U ] being the [ tangent of Psi ] :

  • 

  We then have :

  • 

    based on the [ Tangent Addition Formulas ] we have previously developed , same as above .

  And we take note here that :

  • 

    and

  • 

  Thus , we can now re-write the above equation as :

  • 

  Consequently :

  • 

    yielding :

  • 

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Comparing four (4) Equations :

• Let us now bring in four (4) equations , the first 2 from above , as follows :
  • First Equation :
    • 

  • Second Equation :
    • 

  • Third Equation :
    • 

  • Fourth Equation :
    • 

  And we note here that :

  • the co-efficients are all-the-same throughout but the [ signs ] are different .

• It is appropriate that we also list-out the solution values here for further study :
  • For the Fist Equation :
    • 

  • For the Second Equation :
    • 

  • For the Third Equation :
    • 

  • For the Fourth Equation :
    • 

• Evariste Galois / Felix Klein ?

  Yes , we are looking at Symmetry , and this prompted the next paper :

  • An Approach to the Structural Shape of Polynomial Functions

    due for publication in January , 2015 .

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One Quick Comment here :

• Let us now set up a Fraction :
  • 

  such that :

  • [ M ] and [ N ] are relatively prime ,
    • i.e. :
      • 

      so that the { fraction } cannot be further reduced ;

  • the { fraction } does not evaluate to an integer ,
    • i.e. :
      • 

• We shall now make this statement for further thoughts and evaluation :
  • IF :
    • An angle [ Angle Theta ] is expressible as a { fraction } of { Pi-over-4 } , i.e. :
      • 

        with the { fraction } obeying the special conditions listed immediately above ;

    THEN :

    • the { Tangent of Theta } cannot be a fraction .

  This is because :

  • When [ M = 1 ] , the { tangent of Theta } is the solution to an algebraic equation as discussed above ,
    • and it is most unlikely that this type of algebraic equation can have a [ fraction ] as a solution value ;

  • When [ M ] is { greater that [ 1 ] but less than [ N ] } , we can apply the [ Tangent Addition Fomulas ] as derived previously ,
    • and it is also unlikely that we will come up with a fraction until we reach [ M = N ] .

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go to the next section : Section XLV - A Statement of Purpose for Part VIII

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

return to the MATH Functions Basics HomePage

Original dated 2014-12-07