Trigonometry Angles—π/9

Author

Eric W. Weisstein
October 9, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/TrigonometryAnglesPi9.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TrigonometryAnglesPi9.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Values

Through[{Cos,Sin,Tan}[Pi/9]]

a=With[{n=9},Select[Range[n/2],GCD[#,n]==1&]]

<<Utilities`Simplify`

(values=Flatten[Map[{#,SqrtSimplify[ExpandAll[FullSimplify[RootReduce[Developer`TrigToRadicals[#]]]]]}&,Transpose[Through[{Cos,Cot,Csc,Sec,Sin,Tan}[#Pi/9]]&/@a],
{2}],1])//ColumnForm

Figure

<<Utilities`Plot`

TrigPlot[Pi/9,AnglePosition->{.7,.12}]

Identities

sin((π/9)) sin(((2 π)/9)) sin(((4 π)/9))==((Sqrt[3])/8)

ToRadicals[RootReduce[Simplify[TrigToExp[Sin[(π/9)]Sin[((2π)/9)]Sin[((4π)/9)]]]]]//Timing

Developer`TrigToRadicals[Sin[(π/9)]Sin[((2π)/9)]Sin[((4π)/9)]]//FullSimplify//Timing

cos((π/9)) cos(((2 π)/9)) cos(((4 π)/9))==(1/8)

ToRadicals[RootReduce[Simplify[TrigToExp[Cos[(π/9)]Cos[((2π)/9)]Cos[((4π)/9)]]]]]//Timing

Developer`TrigToRadicals[Cos[(π/9)]Cos[((2π)/9)]Cos[((4π)/9)]]//FullSimplify//Timing

tan((π/9)) tan(((2 π)/9)) tan(((4 π)/9))==(Sqrt[3])

ToRadicals[RootReduce[Simplify[TrigToExp[Tan[(π/9)]Tan[((2π)/9)]Tan[((4π)/9)]]]]]//Timing

Developer`TrigToRadicals[Tan[(π/9)]Tan[((2π)/9)]Tan[((4π)/9)]]//FullSimplify//Timing

Ramanujan's Identity -((cos((π/9)))^(1/3))+((cos(((2 π)/9)))^(1/3))+((cos(((4 π)/9)))^(1/3))==(((3/2) (-2+(3^(2/3))))^(1/3))

RootReduce[Developer`TrigToRadicals[((Cos[((2π)/9)])^(1/3))+((Cos[((4π)/9)])^(1/3))-((Cos[(π/9)])^(1/3))]]

ToRadicals[%]//FullSimplify