Trigonometry Angles—Pi/7

Author

Eric W. Weisstein
October 19, 2021

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/TrigonometryAnglesPi7.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TrigonometryAnglesPi7.html.

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

Figure

<<Utilities`Plot`

TrigPlot[Pi/7,AnglePosition->{.6,.12}]

Values

{HoldForm[#[Pi/7]],RootReduce@Simplify@TrigToExp[#[Pi/7]]}&/@{Cos,Cot,Csc,Sec,Sin,Tan}//TableForm//TraditionalForm

{HoldForm[#[2Pi/7]],RootReduce@Simplify@TrigToExp[#[2Pi/7]]}&/@{Cos,Cot,Csc,Sec,Sin,Tan}//TableForm//TraditionalForm

{HoldForm[#[3Pi/7]],RootReduce@Simplify@TrigToExp[#[3Pi/7]]}&/@{Cos,Cot,Csc,Sec,Sin,Tan}//TableForm//TraditionalForm

Mathematica Syntax

RootReduce[Simplify[TrigToExp[Sin[Pi/7]]]]

V5.2

FunctionExpand[Sin[Pi/7]]

ToRadicals[Sin[Pi/7]]

V6

FunctionExpand[Sin[Pi/7]]

ToRadicals[Sin[Pi/7]]

Developer`TrigToRadicals[Sin[Pi/7]]

V7

FunctionExpand[Sin[Pi/7]]

ToRadicals[Sin[Pi/7]]

Developer`TrigToRadicals[Sin[Pi/7]]

Normalized Forms

cos((π/7))

RootReduce[Simplify[TrigToExp[Cos[Pi/7]]]]

Developer`TrigToRadicals[Cos[Pi/7]]//TraditionalForm

Experimental`OptimizeExpression[Developer`TrigToRadicals[Cos[Pi/7]],OptimizationSymbol->c,OptimizationLevel->2]

sin((π/7))

RootReduce[Simplify[TrigToExp[Sin[Pi/7]]]]

Developer`TrigToRadicals[Sin[Pi/7]]//TraditionalForm

Identities

cos((π/14)) cos(((3 π)/14)) sin((π/7))==((Sqrt[7])/8)

Times@@(Sin[# Pi/7]&/@Range[3])

RootReduce[Simplify[TrigToExp[Cos[(π/14)] Cos[((3 π)/14)] Sin[(π/7)]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Cos[(π/14)] Cos[((3 π)/14)] Sin[(π/7)]]//FullSimplify,60]//Timing

cos((π/7)) sin((π/14)) sin(((3 π)/14))==(1/8)

Times@@(Cos[# Pi/7]&/@Range[3])

RootReduce[Simplify[TrigToExp[Cos[(π/7)] Sin[(π/14)] Sin[((3 π)/14)]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Cos[(π/7)] Sin[(π/14)] Sin[((3 π)/14)]]//FullSimplify,60]//Timing

(cos^2)((π/7))-cos((π/7)) cos(((2 π)/7))==(1/4)

RootReduce[Simplify[TrigToExp[Cos[Pi/7]^2-Cos[Pi/7]Cos[2Pi/7]]]]//Timing

TimeConstrained[Developer`TrigToRadicals[Cos[Pi/7]^2-Cos[Pi/7]Cos[2Pi/7]]//FullSimplify,60]//Timing

sin
𝜋
7

−sin
2𝜋
7

−sin
4𝜋
7

=−7
‾
√2

sin((π/7))-sin(((2 π)/7))-sin(((4 π)/7))==-((Sqrt[7])/2)

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

FullSimplify[Sin[(Pi/7)]-Sin[((2π)/7)]-Sin[((4π)/7)]]

-((cos((π/7)))^(1/3))-((sin((π/14)))^(1/3))+((sin(((3 π)/14)))^(1/3))==-(((1/2) (-5+3 (7^(1/3))))^(1/3))

Ramanujan-like Identity

CubeRoot[Cos[((2π)/7)]]+CubeRoot[Cos[((4π)/7)]]+CubeRoot[Cos[((6π)/7)]]//FullSimplify