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http://functions.wolfram.com/01.06.17.0003.01
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Sin[z] == n Sin[z/n] Product[1 - Sin[z/n]^2/Sin[(4 k Pi)/(2 n)]^2,
{k, 1, Floor[n/2]}] /; Element[(n - 1)/2, Integers] && (n - 1)/2 > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "\[Equal]", RowBox[List["n", " ", RowBox[List["Sin", "[", FractionBox["z", "n"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Sin", "[", FractionBox["z", "n"], "]"]], "2"], SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["4", " ", "k", " ", "\[Pi]"]], RowBox[List["2", " ", "n"]]], "]"]], "2"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sin /> <ci> z </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <sin /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", RowBox[List["Sin", "[", FractionBox["z", "n"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Sin", "[", FractionBox["z", "n"], "]"]], "2"], SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["4", " ", "k", " ", "\[Pi]"]], RowBox[List["2", " ", "n"]]], "]"]], "2"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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