Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Transformations > Some functions of arguments





http://functions.wolfram.com/01.06.16.0163.01









  


  










Input Form





Sin[a (b z^3)^(1/3)] == (-(1/(6 b^(2/3) z^2))) ((I ((-1 + E^(2 I a b^(1/3) z)) (b^(2/3) z^2 + b^(1/3) z (b z^3)^(1/3) + (b z^3)^(2/3)) + E^((1/2) I a b^(1/3) z) ((-b^(1/3)) z + (b z^3)^(1/3)) ((-1 + E^(I a b^(1/3) z)) (2 b^(1/3) z + (b z^3)^(1/3)) Cosh[(1/2) Sqrt[3] a b^(1/3) z] - I Sqrt[3] (1 + E^(I a b^(1/3) z)) (b z^3)^(1/3) Sinh[(1/2) Sqrt[3] a b^(1/3) z])))/E^(I a b^(1/3) z))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["6", " ", SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", "2"]]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["b", RowBox[List["1", "/", "3"]]]]], " ", "z"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["3"], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["3"], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "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> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mroot> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mroot> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 3 </mn> </mroot> <mo> - </mo> <mrow> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mroot> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> b </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", RowBox[List["a_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b_", " ", SuperscriptBox["z_", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["2", "/", "3"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["b", RowBox[List["1", "/", "3"]]]]], " ", "z"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["3"], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "3"]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["3"], " ", "a", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["6", " ", SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", "2"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.