Inverse cosine: Representations through equivalent functions (formula 01.13.27.1220)

ArcCos

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving sinh^-1

Involving cos^-1((c z²)^1/2)

Involving cos^-1((z₂)^1/2) and sinh^-1(i z)

http://functions.wolfram.com/01.13.27.1220.01

Input Form

ArcCos[Sqrt[z^2]] == Pi/2 + ((I Sqrt[z^2])/z) ArcSinh[I z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCos", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", SqrtBox[SuperscriptBox["z", "2"]]]], "z"], RowBox[List["ArcSinh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccos /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", SqrtBox[SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[Pi]", "2"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]], " ", RowBox[List["ArcSinh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "z"]]]]]]]

Date Added to functions.wolfram.com (modification date)

2003-08-21