On the 30-Rhombus Symmetric Solid Object

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

Top Of Page

Section XX - Solving the Equation for the [ Tangent of 9-degrees ]

Summary for the Section :

• In this Section , we shall attempt to solve the equation for the [ Tangent of 9-degrees ] ,
  • developed previously in the last Section .

go to Top Of Page

Bringing-back the Equation :

• Let us now bring back the [ 5th-ordered polynomial equation ] from the last Section :
  • 

  We notice here immediately that :

  • if we substitute [ u = 1 ] into the above equation ,

    the left-hand-side of the equation evaluates to [ 0 ] .

  Consequently , [ u - 1 ] is a factor of the [ 5th-ordered polynominal ] and we have :

  • 

    

• Let us now take a closer look at the [ 4th-ordered polynomial ] on-the-very-right , i.e. :
  • 

  Let us now re-write the said [ 4th-ordered polynomial ] in this special format :

  • 

    

    yielding :

  • 

    

    

  We take note , at this point , that :

  • the right-hand-side of the equation is in the format :
    • [ A-square minus B-square ]

    which is equal to :

    • [ A plus B ] times [ A minus B ] .

• Consequently , based on the original [ 5th-ordered polynominal equation ] , i.e. :
  • 

  we can now write these two (2) Derived Equations , based on the above :

  • First Derived Equation :
    • 

  • Second Derived Equation :
    • 

go to Top Of Page

Solving the First Derived Equation :

• Let us now try to solve the First Derived Equation :
  • 

  We see here that this a Quadratic Equation and the solution is simply :

  • 

  And on expansion , we have :

  • 

    yielding :

  • 

  Consequently :

  • 

    yielding :

  • 

  And the final equation here is then :

  • 

• And on numerical evaluation of the above , we have :
  • 

    and

  • 

go to Top Of Page

Solving the Second Derived Equation :

• Let us now try to solve the Second Derived Equation :
  • 

  We see here that this a Quadratic Equation and the solution is simply :

  • 

  And on expansion , we have :

  • 

    yielding :

  • 

  Consequently :

  • 

    yielding :

  • 

  And the final equation is then :

  • 

• And on numerical evaluation on the above , we have :
  • 

    and

  • 

go to Top Of Page

Summarizing :

• We simply note here that :
  • the [ tangent of 9-degrees ] is indeed [ 0.158384] .

  And we are successful in expressing the [ tangent of 9-degrees ] via [ radicals ] .

go to Top Of Page

go to the next section : Section XXI - Constructing the [ 12-Rhombus Symmetric Solid Object ]

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

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04