On the 30-Rhombus Symmetric Solid Object

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

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Section XIX - Setting up the Equation for the [ Tangent of 9-degrees ]

Summary for the Section :

• In this Section , we shall set up the equation for the [ Tangent of 9-degrees ] .
  • ahead of our solving for the same in the next Section .

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Setting up the Equation :

• Let us now set up :
  • the [ Angle Theta ] as being equal to [ 9-degrees ] , i.e. :
    • 

  and :

  • the unknown variable [ u ] to be calculated as the [ tangent of 9-degrees ] , i.e. :
    • 

• We take note at this point that the [ tangent of 45-degrees ] is equal to [ 1 ] ,
  • and therefore :
    • 

  And based on the formula we have developed in the last section , we can now write :

  • 

    yielding :

  • 

• Consequently :
  • 

    yielding :

  • 

  And this is the [ 5th-ordered polynomial equation ] that we shall be solving for in the next Section .

A Quick Pre-view of the Solution Set :

• Let us just state , at this point , that :
  • the numerical solutions to the above [ 5th-ordered polynominal equation ] are as follows :
    • 

    • 

    • 

    • 

    • 

  And we take note here that :

  • the sequence [ -27 degrees ] / [ 9 degrees ] / [ 45 degrees ] / [ 81 degrees ] / [ 117 degrees ]
    • do differ by a constant [ 36 degrees ] going from-one-item-to-the-next-item .

      

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go to the next section : Section XX - Solving the Equation for the [ Tangent of 9-degrees ]

go to the last section : Section XVIII - Bringing-in the Tangent Addition Formula

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04