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ArcSin




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving tan-1 > Involving sin-1(z1/2/(a+z)1/2) > Involving sin-1(z1/2/(a+z)1/2) and tan-1(z/a1/2)





http://functions.wolfram.com/01.12.27.0051.01









  


  








Input Form





ArcSin[Sqrt[z]/Sqrt[a + z]] == ((Sqrt[z/(a + z)] Sqrt[a + z]) (Pi - Sqrt[a/(a + z)] Sqrt[z/a + 1] (Pi - 2 ArcTan[Sqrt[z/a]])))/ (2 Sqrt[z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["a", "+", "z"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["a", "+", "z"]]]], " ", SqrtBox[RowBox[List["a", "+", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List[SqrtBox[FractionBox["a", RowBox[List["a", "+", "z"]]]], " ", SqrtBox[RowBox[List[FractionBox["z", "a"], "+", "1"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", SqrtBox[FractionBox["z", "a"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["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> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mi> z </mi> </msqrt> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <msqrt> <mfrac> <mi> a </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mi> z </mi> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mi> z </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctan /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", FractionBox[SqrtBox["z_"], SqrtBox[RowBox[List["a_", "+", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["a", "+", "z"]]]], " ", SqrtBox[RowBox[List["a", "+", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List[SqrtBox[FractionBox["a", RowBox[List["a", "+", "z"]]]], " ", SqrtBox[RowBox[List[FractionBox["z", "a"], "+", "1"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", SqrtBox[FractionBox["z", "a"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["z"]]]]]]]]


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





2001-10-29