Trigonometry Angles—Pi/13

Author

Eric W. Weisstein
October 9, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/TrigonometryAnglesPi13.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TrigonometryAnglesPi13.html.

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

Explicit Values

<<Utilities`Typesetting`

RootReduce[FullSimplify[TrigToExp[#[Pi/13]]]]&/@{Cos,Cot,Csc,Sec,Sin,Tan}//TraditionalForm

Expansion of Sin(π/13)

Expand[TrigExpand[Sin[13α]]/.Cos[x_]:>Sqrt[1-Sin[x]^2]]

With[{n=13},(-1)^((n-1)/2)ChebyshevT[n,Sin[α]]]

Factor[Expand[TrigExpand[Sin[13α]]/.Cos[x_]:>Sqrt[1-Sin[x]^2]]]

(13-364 ((Sin[α])^2)+2912 ((Sin[α])^4)-9984 ((Sin[α])^6)+16640 ((Sin[α])^8)-13312 ((Sin[α])^10)+4096 ((Sin[α])^12))/.Sin[α]:>(Sqrt[x])

TeXForm[%]

(s=Developer`TrigToRadicals[Sin[Pi/13]])//TraditionalForm

Identities

<<Utilities`Simplify`

sin((π/13)) sin(((2 π)/13)) sin(((3 π)/13)) sin(((4 π)/13)) sin(((5 π)/13)) sin(((6 π)/13))==((Sqrt[13])/64)

RootReduce[Simplify[TrigToExp[Times@@(Sin[# Pi/13]&/@Range[6])]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Times@@(Sin[# Pi/13]&/@Range[6])]//FullSimplify,60]//Timing

cos((π/13)) cos(((2 π)/13)) cos(((3 π)/13)) cos(((4 π)/13)) cos(((5 π)/13)) cos(((6 π)/13))==(1/64)

RootReduce[Simplify[TrigToExp[Times@@(Cos[# Pi/13]&/@Range[6])]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Times@@(Cos[# Pi/13]&/@Range[6])]//FullSimplify,60]//Timing

(cos^2)((π/13))+(cos^2)(((3 π)/13))+cos(((4 π)/13))==(1/8) (11+(Sqrt[13]))

From: "Pascal Rolli" <pascal.rolli@gmx.net>
Date: Mon, 27 Dec 2004 13:37:03 +0100

RootReduce[Simplify[TrigToExp[((Cos[(π/13)])^2)+((Cos[((3π)/13)])^2)+((Cos[((4π)/13)])^2)]]]//Timing

TimeConstrained[Developer`TrigToRadicals[((Cos[(π/13)])^2)+((Cos[((3π)/13)])^2)+Cos[((4π)/13)]]//FullSimplify,60]//Timing

sin((π/13))+sin(((3 π)/13))+sin(((4 π)/13))==(Sqrt[(13/8)+((3 (Sqrt[13]))/8)])

ToRadicals[RootReduce[Simplify[TrigToExp[Sin[(π/13)]+Sin[((3π)/13)]+Sin[((4π)/13)]]]]]//Timing

Developer`TrigToRadicals[Sin[(π/13)]+Sin[((3π)/13)]+Sin[((4π)/13)]]//FullSimplify//Timing