Inverse Cosine

Author

Eric W. Weisstein
August 24, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/InverseCosine.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

TrigToExp[ArcCos[z]]

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Branch Cut

With[{f=ArcCos},
BranchCutPlot[f,Epilog->Text[TraditionalForm[f[z]],{-2,2.3}]]
]

On Real Axis

Plot[ArcCos[x],{x,-1,1},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,ArcCos[x]},
Epilog->{Red,PointSize[.02],Point/@{{-1,Pi},{1,0}}},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[0,Pi,Pi/4]}]

Complex Plane

ComplexPlot3D[ArcCos[z],{z,-5,5},PlotPoints->100,Mesh->False]//Timing

ComplexContourPlot[ArcCos[z],{z,-5,5}]//Timing

Special Values

ArcCos/@{-1,0,1}

TrigToExp[ArcCos[#]]&/@{-I,I}

Derivative and Integral

Derivative

D[ArcCos[z],z]

Integral

Integrate[ArcCos[z],z]

Identities

In terms of itself

-1≤x≤1

FullSimplify[Through[{Identity,TrigToExp}[π-ArcCos[-z]]]]

FullSimplify[π-ArcCos[-z]-ArcCos[z]]

x≤0

FullSimplify[Pi/2+ArcCos[Sqrt[1-x^2]],x≤0]

x≥0

FullSimplify[Pi/2-ArcCos[Sqrt[1-x^2]],x≥0]

In terms of other arc trigs

Always

FullSimplify[π/2+ArcSin[-z]]

FullSimplify[π/2-ArcSin[z]]

FullSimplify[ArcSec[1/z]]

-1≤x≤1

FullSimplify[π/2-ArcTan[x/Sqrt[1-x^2]],-1<x<1]

x≥0

FullSimplify[ArcCot[x/Sqrt[1-x^2]],x>=0]

FullSimplify[ArcCsc[1/Sqrt[1-x^2]],x>=0]

FullSimplify[ArcSin[Sqrt[1-x^2]],x>=0]

FullSimplify[ArcTan[Sqrt[1-x^2]/x],x>0]

Series

series=Series[ArcCos[x],{x,0,20}]

terms=-Last/@Partition[CoefficientList[series,x],2]

Through[{Numerator,Denominator}[terms]]

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcCos[x],{x,0,n}]

...wrong...

V7

SeriesCoefficient[ArcCos[x],{x,0,n}]

Sum

(π/2)-(∑_n^∞)(((Subscript[((1/2)), n-1]) (x^(2 n-1)))/((2 n-1) (n-1)!))==(cos^(-1))(x)

(π/2)-Sum[((Pochhammer[1/2,n-1])/((n-1)!(2n-1)))(x^(2n-1)),{n,∞}]

FullSimplify[%]

(∑_n^∞)(((x^n) (Γ((n/2)) θ(n-1) (Subscript[δ, (n-1) mod 2])))/((Sqrt[π]) n Γ(((n+1)/2))))==(sin^(-1))(x)

Sum[((Gamma[(n/2)] KroneckerDelta[Mod[-1+n,2]] UnitStep[-1+n])/(n (Sqrt[π]) Gamma[((1+n)/2)]))(x^n),{n,∞}]

With[{x=.8},{Sum[-((Gamma[(n/2)] KroneckerDelta[Mod[-1+n,2]] UnitStep[-1+n])/(n (Sqrt[π]) Gamma[((1+n)/2)]))(x^n),{n,10}],ArcSin[x]}]

ArcSin[.2]