Trigonometry Angles—Pi/11

Author

Eric W. Weisstein
October 19, 2021

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/TrigonometryAnglesPi11.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TrigonometryAnglesPi11.html.

©2021 Wolfram Research, Inc. except for portions noted otherwise

Explicit Values

<<Utilities`Typesetting`

RootReduce[FullSimplify[TrigToExp[#[Pi/11]]]]&/@{Cos,Cot,Csc,Sec,Sin,Tan}//TraditionalForm

Expansion of Sin(π/11)

Expand[TrigExpand[Sin[11α]]/.Cos[x_]:>Sqrt[1-Sin[x]^2]]

With[{n=11},(-1)^((n-1)/2)ChebyshevT[n,Sin[α]]]

(s=Developer`TrigToRadicals[Sin[Pi/11]])//TraditionalForm

Chop[N[s,20]]

N[Sin[Pi/11],20]

Identities

<<Utitilities`Simplify`

sin((π/11)) sin(((2 π)/11)) sin(((3 π)/11)) sin(((4 π)/11)) sin(((5 π)/11))==((Sqrt[11])/32)

Times@@(Sin[# Pi/11]&/@Range[5])

ToRadicals[RootReduce[Simplify[TrigToExp[Cos[(π/22)] Cos[((3 π)/22)] Cos[((5 π)/22)] Sin[(π/11)] Sin[((2 π)/11)]]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Cos[(π/22)] Cos[((3 π)/22)] Cos[((5 π)/22)] Sin[(π/11)] Sin[((2 π)/11)]]//FullSimplify,60]//Timing

cos((π/11)) cos(((2 π)/11)) cos(((3 π)/11)) cos(((4 π)/11)) cos(((5 π)/11))==(1/32)

Times@@(Cos[# Pi/11]&/@Range[5])

ToRadicals[RootReduce[Simplify[TrigToExp[Cos[(π/11)] Cos[((2 π)/11)] Sin[(π/22)] Sin[((3 π)/22)] Sin[((5 π)/22)]]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Cos[(π/11)] Cos[((2 π)/11)] Sin[(π/22)] Sin[((3 π)/22)] Sin[((5 π)/22)]]//FullSimplify,60]//Timing

4 sin(((2 π)/11))+tan(((3 π)/11))==(Sqrt[11])

ToRadicals[RootReduce[Simplify[TrigToExp[Tan[((3π)/11)]+4Sin[((2π)/11)]]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Tan[((3π)/11)]+4Sin[((2π)/11)]]//FullSimplify,60]//Timing

sin((π/11))+sin(((2 π)/11))+cos((π/22))-cos(((3 π)/22))+cos(((5 π)/22))==((Sqrt[11])/2)
-sin((π/22))+sin(((3 π)/22))-sin(((5 π)/22))-cos((π/11))+cos(((2 π)/11))==-(1/2)

https://math.stackexchange.com/questions/4277731/on-the-representation-of-sqrt-pm-p-in-the-integral-basis-of-mathbb-q-zet

With[{z=Exp[((2Pi I)/11)]},z+(z^3)+(z^4)+(z^5)+(z^9)]//FullSimplify//Expand

Re[z+(z^3)+(z^4)+(z^5)+(z^9)/.z->Exp[((2Pi I)/11)]]

FullSimplify[%]

Im[z+(z^3)+(z^4)+(z^5)+(z^9)/.z->Exp[((2Pi I)/11)]]

FullSimplify[%]