van Roomen's Equation

Author

Eric W. Weisstein
August 28, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Algebra/vanRoomensEquation.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Equation

lhs1=2ChebyshevT[45,(x/2)]//Expand

lhs2=FunctionExpand[2Sin[45θ]/.θ->ArcSin[(x/2)]]

lhs1==lhs2

b1=(Sqrt[(7/4)-((Sqrt[5])/4)-(Sqrt[(15/8)-((3 (Sqrt[5]))/8)])]);

b2=2Sin[(π/15)];

RootReduce[b1-b2]

roots=x/.{ToRules[Reduce[lhs1==b1,x]]}

Length[roots]

posroots=x/.{ToRules[Reduce[lhs1==b1,x,PositiveReals]]}

Length[posroots]

N[posroots[[1]]-2Sin[(π/675)],100]//Quiet

FullSimplify[posroots[[1]]-2Sin[(π/675)]]

RootReduce[posroots[[1]]-2Sin[(π/675)]]//Timing