Half-Angle Formulas

Author

Eric W. Weisstein
January 4, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/Half-AngleFormulas.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identities

sin((x/2))=(Sqrt[((1-cos x)/2)])

Simplify[Subtract@@{Sin[x/2],Sqrt[(1-Cos[x])/2]},0<x<2Pi]

FullSimplify[Subtract@@{Sin[x/2],(-1)^Floor[(x/(2π))](Sqrt[((1-Cos[x])/2)])},x∈Reals]

Plot[Sin[x/2]-(-1)^Floor[(x/(2π))](Sqrt[((1-Cos[x])/2)]),{x,-7π,7π},PlotStyle->Red,PlotRange->All]

cos((x/2))=-(Sqrt[ ((cos x+1)/2)])

Simplify[Subtract@@{Cos[x/2],Sqrt[(1+Cos[x])/2]},0<x<Pi]

Simplify[Subtract@@{Cos[x/2],-Sqrt[(1+Cos[x])/2]},Pi<x<2Pi]

FullSimplify[Subtract@@{Cos[x/2],(-1)^Floor[((x+Pi)/(2π))](Sqrt[((1+Cos[x])/2)])},x∈Reals]

Plot[Cos[x/2]-(-1)^(Floor[((x+Pi)/(2π))])(Sqrt[((1+Cos[x])/2)]),{x,-30,30},PlotStyle->Red,PlotRange->All]

Tan[x/2]

Simplify[Tan[x/2]-((Sin[x])/(1+Cos[x]))]

Simplify[Tan[x/2]-((1-Cos[x])/(Sin[x]))]

Simplify[Tan[x/2]-((Tan[x]Sin[x])/(Tan[x]+Sin[x]))]

Simplify[Tan[x/2]-(((Sqrt[((Tan[x])^2)+1])-1)/(Tan[x])),-π/2<x<π/2]

Plot[Tan[x/2]-(((((-1))^(Floor[(x+Pi/2)/(Pi)]))(Sqrt[((Tan[x])^2)+1])-1)/(Tan[x])),{x,-20,20},PlotStyle->Red,PlotRange->All]

Plot[{Tan[x/2],Sqrt[(1-Cos[x])/(1+Cos[x])]},{x,0,2π},PlotStyle->{Red,Blue}]

FullSimplify[Subtract@@{Tan[x/2],(((-1))^(Floor[x/Pi]))(Sqrt[((1-Cos[x])/(1+Cos[x]))])},x∈Reals]

Plot[Tan[x/2]-(((-1))^(Floor[x/Pi]))(Sqrt[((1-Cos[x])/(1+Cos[x]))]),{x,-30,30},PlotStyle->Red,PlotPoints->100,PlotRange->All]

Simplify[Subtract@@{Tan[x/2],Sqrt[(1-Cos[x])/(1+Cos[x])]},0<x<Pi]

FullSimplify[Subtract@@{Tan[x/2],-Sqrt[(1-Cos[x])/(1+Cos[x])]},Pi<x<2Pi]

FullSimplify[Subtract@@{Tan[x/2],-Sqrt[(1-Cos[x])/(1+Cos[x])]},Pi<x<2Pi]

Plot[Tan[x],{x,0,2π}]

Simplify[Abs[Tan[x]],0<x<Pi/2]

Simplify[Abs[Tan[x]],Pi/2<x<Pi]

FullSimplify[Abs[Tan[x]],Pi<x<3Pi/2]

Refine[Abs[Tan[x]],3Pi/2<x<2Pi]

FullSimplify[TrigToExp[Abs[Tan[x]]],Pi<x<3Pi/2]

Plot[Tan[x/2]-(((Sqrt[((Tan[x])^2)+1])-1)/(Tan[x])),{x,-30,30},PlotStyle->Red,PlotPoints->100,PlotRange->All]

Function

For 0 < A, B, C < π

Plot[Sec[x/2],{x,0,Pi},PlotStyle->Red]

Expand//@Simplify[DistanceTrilinears[HalfAngleFormulas[Sec[#/2]]]]&/@{A,B,C}

Plot[Cos[x/2],{x,0,Pi},PlotStyle->Red]

Together//@Simplify[DistanceTrilinears[HalfAngleFormulas[Cos[#/2]]]]&/@{A,B,C}

Together//@Simplify[DistanceTrilinears[HalfAngleFormulas[Sec[#/2]]]]&/@{A,B,C}

HalfAngleFormulas[expr_]:=expr/.{
Csc[x_/2]^n_.:>(2/(1-Cos[x]))^(n/2),
Cos[x_/2]^n_.:>((1+Cos[x])/2)^(n/2),
Sec[x_/2]^n_.:>(2/(1+Cos[x]))^(n/2),
Sin[x_/2]^n_.:>((1-Cos[x])/2)^(n/2),
Tan[x_/2]:>(1-Cos[x])/Sin[x]}

Together//@Simplify[DistanceTrilinears[HalfAngleFormulas[Sin[#/2]]]]&/@{A,B,C}

Together//@Simplify[DistanceTrilinears[HalfAngleFormulas[Csc[#/2]]]]&/@{A,B,C}

LawOfHalfCosines[expr_]:=expr/.{
 (Sqrt[((-(a^2)+(b^2)+2 b c+(c^2))/(b c))]):>2Cos[A/2], 
(Sqrt[(((a^2)-(b^2)+2 a c+(c^2))/(a c))]):>2Cos[B/2], 
(Sqrt[(((a^2)+2 a b+(b^2)-(c^2))/(a b))]):>2Cos[C/2],
 (Sqrt[((b c)/(-(a^2)+(b^2)+2 b c+(c^2)))]):>Sec[A/2]/2,
(Sqrt[((a c)/((a^2)-(b^2)+2 a c+(c^2)))]):>Sec[B/2]/2,
(Sqrt[((a b)/((a^2)+2 a b+(b^2)-(c^2)))]):>Sec[C/2]/2, 
(Sqrt[(((a^2)-(b^2)+2 b c-(c^2))/(b c))]):>2Sin[A/2],
 (Sqrt[((-(a^2)+(b^2)+2 a c-(c^2))/(a c))]):>2Sin[B/2],
 (Sqrt[((-(a^2)+2 a b-(b^2)+(c^2))/(a b))]):>2Sin[C/2],
 (Sqrt[-((b c)/(-(a^2)+(b^2)-2 b c+(c^2)))]):>Csc[A/2]/2,
 (Sqrt[-((a c)/((a^2)-(b^2)-2 a c+(c^2)))]):>Csc[B/2]/2,
 (Sqrt[-((a b)/((a^2)-2 a b+(b^2)-(c^2)))]):>Csc[C/2]/2
}

LawOfHalfCosines[expr_] := expr /. {
	Sqrt[(-a^2 + b^2 + 2*b*c + c^2)/(b*c)] :> 2*Cos[A/2], 
    Sqrt[(a^2 - b^2 + 2*a*c + c^2)/(a*c)] :> 2*Cos[B/2], 
    Sqrt[(a^2 + 2*a*b + b^2 - c^2)/(a*b)] :> 2*Cos[C/2], 
    Sqrt[(b*c)/(-a^2 + b^2 + 2*b*c + c^2)] :> Sec[A/2]/2, 
    Sqrt[(a*c)/(a^2 - b^2 + 2*a*c + c^2)] :> Sec[B/2]/2, 
    Sqrt[(a*b)/(a^2 + 2*a*b + b^2 - c^2)] :> Sec[C/2]/2, 
    Sqrt[(a^2 - b^2 + 2*b*c - c^2)/(b*c)] :> 2*Sin[A/2], 
    Sqrt[(-a^2 + b^2 + 2*a*c - c^2)/(a*c)] :> 2*Sin[B/2], 
    Sqrt[(-a^2 + 2*a*b - b^2 + c^2)/(a*b)] :> 2*Sin[C/2], 
    Sqrt[-((b*c)/(-a^2 + b^2 - 2*b*c + c^2))] :> Csc[A/2]/2, 
    Sqrt[-((a*c)/(a^2 - b^2 - 2*a*c + c^2))] :> Csc[B/2]/2, 
    Sqrt[-((a*b)/(a^2 - 2*a*b + b^2 - c^2))] :> Csc[C/2]/2}