<?xml version='1.0'?> <?xml-stylesheet type='text/xsl' href='pmathml.xsl'?> <html xmlns='http://www.w3.org/1999/xhtml'> <head> <title>Trigonometric and Hyperbolic Sine and Cosine Reverse Theory</title> <meta name="description" id="description" content="Trigonometric and Hyperbolic Sine and Cosine Reverse Theory"/> <meta name="keywords" id="keywords" content=" sin reverse sinh cos cosh theory "/> <style type='text/css'> body { color : black } body { background-color : white } A:link { color : blue } A:visited { color : purple } A:active { color : purple } </style> <script type='text/javascript' language='JavaScript' src='_sincosreverse_xml.js'> </script> </head> <body> <table><tr> <td> <a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a> </td> <td><a href="sqrtreverse.xml" target="_top">Prev</a> </td><td><a href="atanreverse.xml" target="_top">Next</a> </td><td> <select onchange='choose_across0(this)'> <option>Index-></option> <option>contents</option> <option>reference</option> <option>index</option> <option>search</option> <option>external</option> </select> </td> <td> <select onchange='choose_up0(this)'> <option>Up-></option> <option>CppAD</option> <option>Appendix</option> <option>Theory</option> <option>ReverseTheory</option> <option>SinCosReverse</option> </select> </td> <td> <select onchange='choose_down3(this)'> <option>Appendix-></option> <option>Faq</option> <option>speed</option> <option>Theory</option> <option>glossary</option> <option>Bib</option> <option>Bugs</option> <option>WishList</option> <option>whats_new</option> <option>include_deprecated</option> <option>License</option> </select> </td> <td> <select onchange='choose_down2(this)'> <option>Theory-></option> <option>ForwardTheory</option> <option>ReverseTheory</option> <option>reverse_identity</option> </select> </td> <td> <select onchange='choose_down1(this)'> <option>ReverseTheory-></option> <option>ExpReverse</option> <option>LogReverse</option> <option>SqrtReverse</option> <option>SinCosReverse</option> <option>AtanReverse</option> <option>AsinReverse</option> <option>AcosReverse</option> </select> </td> <td>SinCosReverse</td> <td>Headings</td> </tr></table><br/> <center><b><big><big>Trigonometric and Hyperbolic Sine and Cosine Reverse Theory</big></big></b></center> We use the reverse theory <a href="reversetheory.xml#Standard Math Functions" target="_top"><span style='white-space: nowrap'>standard math function</span></a> definition for the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>H</mi> </mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>G</mi> </mrow></math> . In addition, we use the following definitions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>s</mi> </mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>c</mi> </mrow></math> and the integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mo stretchy="false">ℓ</mo> </mrow></math> <table><tr><td align='left' valign='top'> Coefficients </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>s</mi> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>c</mi> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mo stretchy="false">ℓ</mo> </mrow></math> </td></tr><tr><td align='left' valign='top'> Trigonometric Case </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi>sin</mi> <mo stretchy="false">[</mo> <mi mathvariant='italic'>X</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi>cos</mi> <mo stretchy="false">[</mo> <mi mathvariant='italic'>X</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> 1 </td></tr><tr><td align='left' valign='top'> Hyperbolic Case </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi>sinh</mi> <mo stretchy="false">[</mo> <mi mathvariant='italic'>X</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi>cosh</mi> <mo stretchy="false">[</mo> <mi mathvariant='italic'>X</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> -1 </td></tr> </table> We use the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <msup><mi mathvariant='italic'>z</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">=</mo> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">,</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow></math> in the definition for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>G</mi> </mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>H</mi> </mrow></math> . The forward mode formulas for the <a href="sincosforward.xml" target="_top"><span style='white-space: nowrap'>sine and cosine</span></a> functions are <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mtable rowalign="center" ><mtr><mtd columnalign="right" > <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mn>1</mn> <mo stretchy="false">+</mo> <mo stretchy="false">ℓ</mo> </mrow> <mrow><mn>2</mn> </mrow> </mfrac> <mi>sin</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">+</mo> <mfrac><mrow><mn>1</mn> <mo stretchy="false">-</mo> <mo stretchy="false">ℓ</mo> </mrow> <mrow><mn>2</mn> </mrow> </mfrac> <mi>sinh</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mtd></mtr><mtr><mtd columnalign="right" > <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mn>1</mn> <mo stretchy="false">+</mo> <mo stretchy="false">ℓ</mo> </mrow> <mrow><mn>2</mn> </mrow> </mfrac> <mi>cos</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">+</mo> <mfrac><mrow><mn>1</mn> <mo stretchy="false">-</mo> <mo stretchy="false">ℓ</mo> </mrow> <mrow><mn>2</mn> </mrow> </mfrac> <mi>cosh</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mtd></mtr></mtable> </mrow></math> for the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>j</mi> <mo stretchy="false">=</mo> <mn>0</mn> </mrow></math> , and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>j</mi> <mo stretchy="false">></mo> <mn>0</mn> </mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mtable rowalign="center" ><mtr><mtd columnalign="right" > <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </mfrac> <munderover><mo displaystyle='true' largeop='true'>∑</mo> <mrow><mi mathvariant='italic'>k</mi> <mo stretchy="false">=</mo> <mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </munderover> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr><mtr><mtd columnalign="right" > <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mo stretchy="false">ℓ</mo> <mfrac><mrow><mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </mfrac> <munderover><mo displaystyle='true' largeop='true'>∑</mo> <mrow><mi mathvariant='italic'>k</mi> <mo stretchy="false">=</mo> <mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </munderover> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr></mtable> </mrow></math> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>j</mi> <mo stretchy="false">=</mo> <mn>0</mn> </mrow></math> , we have the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mtable rowalign="center" ><mtr><mtd columnalign="right" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>H</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mo stretchy="false">+</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">+</mo> <mo stretchy="false">ℓ</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr></mtable> </mrow></math> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>j</mi> <mo stretchy="false">></mo> <mn>0</mn> </mrow></math> , then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>k</mi> <mo stretchy="false">=</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mo stretchy="false">…</mo> <mo stretchy="false">,</mo> <mi mathvariant='italic'>j</mi> <mn>-1</mn> </mrow></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mtable rowalign="center" ><mtr><mtd columnalign="right" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>H</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mo stretchy="false">+</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mfrac><mrow><mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </mfrac> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">+</mo> <mo stretchy="false">ℓ</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mfrac><mrow><mn>1</mn> </mrow> <mrow><mi mathvariant='italic'>j</mi> </mrow> </mfrac> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr><mtr><mtd columnalign="right" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>H</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mo stretchy="false">+</mo> <mo stretchy="false">ℓ</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr><mtr><mtd columnalign="right" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>H</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>c</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">-</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mo stretchy="false">+</mo> <mfrac><mrow><mo stretchy="false">∂</mo> <mi mathvariant='italic'>G</mi> </mrow> <mrow><mo stretchy="false">∂</mo> <msup><mi mathvariant='italic'>s</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> <mi mathvariant='italic'>k</mi> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd></mtr></mtable> </mrow></math> <hr/>Input File: omh/sin_cos_reverse.omh </body> </html>