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<a name="Trigonometric"></a> |
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<a name="SEC51"></a> |
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<table cellpadding="1" cellspacing="1" border="0"> |
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<tr><td valign="middle" align="left">[<a href="maxima_14.html#SEC50" title="Previous section in reading order"> < </a>]</td> |
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<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td> |
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</tr></table> |
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<h1 class="chapter"> 15. Trigonometric </h1> |
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<table class="menu" border="0" cellspacing="0"> |
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<tr><td align="left" valign="top"><a href="#SEC52">15.1 Introduction to Trigonometric</a></td><td> </td><td align="left" valign="top"> |
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</td></tr> |
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<tr><td align="left" valign="top"><a href="#SEC53">15.2 Definitions for Trigonometric</a></td><td> </td><td align="left" valign="top"> |
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</td></tr> |
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</table> |
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<hr size="6"> |
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<a name="Introduction-to-Trigonometric"></a> |
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<a name="SEC52"></a> |
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<table cellpadding="1" cellspacing="1" border="0"> |
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<tr><td valign="middle" align="left">[<a href="#SEC51" title="Previous section in reading order"> < </a>]</td> |
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<td valign="middle" align="left">[<a href="#SEC51" title="Beginning of this chapter or previous chapter"> << </a>]</td> |
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<td valign="middle" align="left">[<a href="#SEC51" title="Up section"> Up </a>]</td> |
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<td valign="middle" align="left"> </td> |
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<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td> |
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</tr></table> |
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<h2 class="section"> 15.1 Introduction to Trigonometric </h2> |
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<p>Maxima has many trigonometric functions defined. Not all trigonometric |
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identities are programmed, but it is possible for the user to add many |
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of them using the pattern matching capabilities of the system. The |
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trigonometric functions defined in Maxima are: <code>acos</code>, |
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<code>acosh</code>, <code>acot</code>, <code>acoth</code>, <code>acsc</code>, |
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<code>acsch</code>, <code>asec</code>, <code>asech</code>, <code>asin</code>, |
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<code>asinh</code>, <code>atan</code>, <code>atanh</code>, <code>cos</code>, |
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<code>cosh</code>, <code>cot</code>, <code>coth</code>, <code>csc</code>, <code>csch</code>, |
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<code>sec</code>, <code>sech</code>, <code>sin</code>, <code>sinh</code>, <code>tan</code>, |
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and <code>tanh</code>. There are a number of commands especially for |
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handling trigonometric functions, see <code>trigexpand</code>, |
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<code>trigreduce</code>, and the switch <code>trigsign</code>. Two share |
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packages extend the simplification rules built into Maxima, |
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<code>ntrig</code> and <code>atrig1</code>. Do <code>describe(<var>command</var>)</code> |
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for details. |
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</p> |
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<hr size="6"> |
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<a name="Definitions-for-Trigonometric"></a> |
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<a name="SEC53"></a> |
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<table cellpadding="1" cellspacing="1" border="0"> |
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<tr><td valign="middle" align="left">[<a href="#SEC52" title="Previous section in reading order"> < </a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_16.html#SEC54" title="Next section in reading order"> > </a>]</td> |
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<td valign="middle" align="left"> </td> |
|
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<td valign="middle" align="left">[<a href="#SEC51" title="Beginning of this chapter or previous chapter"> << </a>]</td> |
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<td valign="middle" align="left">[<a href="#SEC51" title="Up section"> Up </a>]</td> |
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<td valign="middle" align="left">[<a href="maxima_16.html#SEC54" title="Next chapter"> >> </a>]</td> |
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<td valign="middle" align="left"> </td> |
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<td valign="middle" align="left"> </td> |
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<td valign="middle" align="left"> </td> |
|
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<td valign="middle" align="left"> </td> |
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<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td> |
|
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<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td> |
|
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<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td> |
|
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<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td> |
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</tr></table> |
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<h2 class="section"> 15.2 Definitions for Trigonometric </h2> |
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<dl> |
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<dt><u>Function:</u> <b>acos</b><i> (<var>x</var>)</i> |
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<a name="IDX475"></a> |
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</dt> |
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<dd><p> - Arc Cosine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>acosh</b><i> (<var>x</var>)</i> |
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<a name="IDX476"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Cosine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>acot</b><i> (<var>x</var>)</i> |
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<a name="IDX477"></a> |
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</dt> |
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<dd><p> - Arc Cotangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>acoth</b><i> (<var>x</var>)</i> |
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<a name="IDX478"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Cotangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>acsc</b><i> (<var>x</var>)</i> |
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<a name="IDX479"></a> |
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</dt> |
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<dd><p> - Arc Cosecant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>acsch</b><i> (<var>x</var>)</i> |
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<a name="IDX480"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Cosecant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>asec</b><i> (<var>x</var>)</i> |
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<a name="IDX481"></a> |
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</dt> |
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<dd><p> - Arc Secant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>asech</b><i> (<var>x</var>)</i> |
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<a name="IDX482"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Secant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>asin</b><i> (<var>x</var>)</i> |
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<a name="IDX483"></a> |
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</dt> |
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<dd><p> - Arc Sine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>asinh</b><i> (<var>x</var>)</i> |
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<a name="IDX484"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Sine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>atan</b><i> (<var>x</var>)</i> |
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<a name="IDX485"></a> |
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</dt> |
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<dd><p> - Arc Tangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>atan2</b><i> (<var>y</var>, <var>x</var>)</i> |
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<a name="IDX486"></a> |
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</dt> |
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<dd><p>- yields the value of <code>atan(<var>y</var>/<var>x</var>)</code> in the interval <code>-%pi</code> to |
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<code>%pi</code>. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>atanh</b><i> (<var>x</var>)</i> |
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<a name="IDX487"></a> |
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</dt> |
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<dd><p> - Hyperbolic Arc Tangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Package:</u> <b>atrig1</b> |
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<a name="IDX488"></a> |
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</dt> |
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<dd><p>The <code>atrig1</code> package contains several additional simplification rules |
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for inverse trigonometric functions. Together with rules |
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already known to Maxima, the following angles are fully implemented: |
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<code>0</code>, <code>%pi/6</code>, <code>%pi/4</code>, <code>%pi/3</code>, and <code>%pi/2</code>. |
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Corresponding angles in the other three quadrants are also available. |
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Do <code>load(atrig1);</code> to use them. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>cos</b><i> (<var>x</var>)</i> |
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<a name="IDX489"></a> |
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</dt> |
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<dd><p> - Cosine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>cosh</b><i> (<var>x</var>)</i> |
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<a name="IDX490"></a> |
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</dt> |
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<dd><p> - Hyperbolic Cosine. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>cot</b><i> (<var>x</var>)</i> |
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<a name="IDX491"></a> |
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</dt> |
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<dd><p> - Cotangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>coth</b><i> (<var>x</var>)</i> |
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<a name="IDX492"></a> |
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</dt> |
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<dd><p> - Hyperbolic Cotangent. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>csc</b><i> (<var>x</var>)</i> |
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<a name="IDX493"></a> |
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</dt> |
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<dd><p> - Cosecant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Function:</u> <b>csch</b><i> (<var>x</var>)</i> |
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<a name="IDX494"></a> |
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</dt> |
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<dd><p> - Hyperbolic Cosecant. |
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</p> |
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</dd></dl> |
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<dl> |
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<dt><u>Option variable:</u> <b>halfangles</b> |
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<a name="IDX495"></a> |
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</dt> |
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<dd><p>Default value: <code>false</code> |
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</p> |
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<p>When <code>halfangles</code> is <code>true</code>, |
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half-angles are simplified away. |
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</p> |
|
392 |
</dd></dl> |
|
393 |
||
394 |
<dl> |
|
395 |
<dt><u>Package:</u> <b>ntrig</b> |
|
396 |
<a name="IDX496"></a> |
|
397 |
</dt> |
|
398 |
<dd><p>The <code>ntrig</code> package contains a set of simplification rules that are |
|
399 |
used to simplify trigonometric function whose arguments are of the form |
|
400 |
<code><var>f</var>(<var>n</var> %pi/10)</code> where <var>f</var> is any of the functions |
|
401 |
<code>sin</code>, <code>cos</code>, <code>tan</code>, <code>csc</code>, <code>sec</code> and <code>cot</code>. |
|
402 |
</p> |
|
403 |
</dd></dl> |
|
404 |
||
405 |
<dl> |
|
406 |
<dt><u>Function:</u> <b>sec</b><i> (<var>x</var>)</i> |
|
407 |
<a name="IDX497"></a> |
|
408 |
</dt> |
|
409 |
<dd><p> - Secant. |
|
410 |
</p> |
|
411 |
</dd></dl> |
|
412 |
||
413 |
<dl> |
|
414 |
<dt><u>Function:</u> <b>sech</b><i> (<var>x</var>)</i> |
|
415 |
<a name="IDX498"></a> |
|
416 |
</dt> |
|
417 |
<dd><p> - Hyperbolic Secant. |
|
418 |
</p> |
|
419 |
</dd></dl> |
|
420 |
||
421 |
<dl> |
|
422 |
<dt><u>Function:</u> <b>sin</b><i> (<var>x</var>)</i> |
|
423 |
<a name="IDX499"></a> |
|
424 |
</dt> |
|
425 |
<dd><p> - Sine. |
|
426 |
</p> |
|
427 |
</dd></dl> |
|
428 |
||
429 |
<dl> |
|
430 |
<dt><u>Function:</u> <b>sinh</b><i> (<var>x</var>)</i> |
|
431 |
<a name="IDX500"></a> |
|
432 |
</dt> |
|
433 |
<dd><p> - Hyperbolic Sine. |
|
434 |
</p> |
|
435 |
</dd></dl> |
|
436 |
||
437 |
<dl> |
|
438 |
<dt><u>Function:</u> <b>tan</b><i> (<var>x</var>)</i> |
|
439 |
<a name="IDX501"></a> |
|
440 |
</dt> |
|
441 |
<dd><p> - Tangent. |
|
442 |
</p> |
|
443 |
</dd></dl> |
|
444 |
||
445 |
<dl> |
|
446 |
<dt><u>Function:</u> <b>tanh</b><i> (<var>x</var>)</i> |
|
447 |
<a name="IDX502"></a> |
|
448 |
</dt> |
|
449 |
<dd><p> - Hyperbolic Tangent. |
|
450 |
</p> |
|
451 |
</dd></dl> |
|
452 |
||
453 |
<dl> |
|
454 |
<dt><u>Function:</u> <b>trigexpand</b><i> (<var>expr</var>)</i> |
|
455 |
<a name="IDX503"></a> |
|
456 |
</dt> |
|
457 |
<dd><p>Expands trigonometric and hyperbolic functions of |
|
458 |
sums of angles and of multiple angles occurring in <var>expr</var>. For best |
|
459 |
results, <var>expr</var> should be expanded. To enhance user control of |
|
460 |
simplification, this function expands only one level at a time, |
|
461 |
expanding sums of angles or multiple angles. To obtain full expansion |
|
462 |
into sines and cosines immediately, set the switch <code>trigexpand: true</code>. |
|
463 |
</p> |
|
464 |
<p><code>trigexpand</code> is governed by the following global flags: |
|
465 |
</p> |
|
466 |
<dl compact="compact"> |
|
467 |
<dt> <code>trigexpand</code></dt> |
|
468 |
<dd><p>If <code>true</code> causes expansion of all |
|
469 |
expressions containing sin's and cos's occurring subsequently. |
|
470 |
</p></dd> |
|
471 |
<dt> <code>halfangles</code></dt> |
|
472 |
<dd><p>If <code>true</code> causes half-angles to be simplified |
|
473 |
away. |
|
474 |
</p></dd> |
|
475 |
<dt> <code>trigexpandplus</code></dt> |
|
476 |
<dd><p>Controls the "sum" rule for <code>trigexpand</code>, |
|
477 |
expansion of sums (e.g. <code>sin(x + y)</code>) will take place only if |
|
478 |
<code>trigexpandplus</code> is <code>true</code>. |
|
479 |
</p></dd> |
|
480 |
<dt> <code>trigexpandtimes</code></dt> |
|
481 |
<dd><p>Controls the "product" rule for <code>trigexpand</code>, |
|
482 |
expansion of products (e.g. <code>sin(2 x)</code>) will take place only if |
|
483 |
<code>trigexpandtimes</code> is <code>true</code>. |
|
484 |
</p></dd> |
|
485 |
</dl> |
|
486 |
||
487 |
<p>Examples: |
|
488 |
</p> |
|
489 |
<table><tr><td> </td><td><pre class="example">(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand; |
|
490 |
2 2 |
|
491 |
(%o1) - sin (x) + 3 cos (x) + x |
|
492 |
(%i2) trigexpand(sin(10*x+y)); |
|
493 |
(%o2) cos(10 x) sin(y) + sin(10 x) cos(y) |
|
494 |
||
495 |
</pre></td></tr></table> |
|
496 |
</dd></dl> |
|
497 |
||
498 |
<dl> |
|
499 |
<dt><u>Option variable:</u> <b>trigexpandplus</b> |
|
500 |
<a name="IDX504"></a> |
|
501 |
</dt> |
|
502 |
<dd><p>Default value: <code>true</code> |
|
503 |
</p> |
|
504 |
<p><code>trigexpandplus</code> controls the "sum" rule for |
|
505 |
<code>trigexpand</code>. Thus, when the <code>trigexpand</code> command is used or the |
|
506 |
<code>trigexpand</code> switch set to <code>true</code>, expansion of sums |
|
507 |
(e.g. <code>sin(x+y))</code> will take place only if <code>trigexpandplus</code> is |
|
508 |
<code>true</code>. |
|
509 |
</p> |
|
510 |
</dd></dl> |
|
511 |
||
512 |
<dl> |
|
513 |
<dt><u>Option variable:</u> <b>trigexpandtimes</b> |
|
514 |
<a name="IDX505"></a> |
|
515 |
</dt> |
|
516 |
<dd><p>Default value: <code>true</code> |
|
517 |
</p> |
|
518 |
<p><code>trigexpandtimes</code> controls the "product" rule for |
|
519 |
<code>trigexpand</code>. Thus, when the <code>trigexpand</code> command is used or the |
|
520 |
<code>trigexpand</code> switch set to <code>true</code>, expansion of products (e.g. <code>sin(2*x)</code>) |
|
521 |
will take place only if <code>trigexpandtimes</code> is <code>true</code>. |
|
522 |
</p> |
|
523 |
</dd></dl> |
|
524 |
||
525 |
<dl> |
|
526 |
<dt><u>Option variable:</u> <b>triginverses</b> |
|
527 |
<a name="IDX506"></a> |
|
528 |
</dt> |
|
529 |
<dd><p>Default value: <code>all</code> |
|
530 |
</p> |
|
531 |
<p><code>triginverses</code> controls the simplification of the |
|
532 |
composition of trigonometric and hyperbolic functions with their inverse |
|
533 |
functions. |
|
534 |
</p> |
|
535 |
<p>If <code>all</code>, both e.g. <code>atan(tan(<var>x</var>))</code> |
|
536 |
and <code>tan(atan(<var>x</var>))</code> simplify to <var>x</var>. |
|
537 |
</p> |
|
538 |
<p>If <code>true</code>, the <code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code> |
|
539 |
simplification is turned off. |
|
540 |
</p> |
|
541 |
<p>If <code>false</code>, both the |
|
542 |
<code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code> and |
|
543 |
<code><var>fun</var>(<var>arcfun</var>(<var>x</var>))</code> |
|
544 |
simplifications are turned off. |
|
545 |
</p> |
|
546 |
</dd></dl> |
|
547 |
||
548 |
<dl> |
|
549 |
<dt><u>Function:</u> <b>trigreduce</b><i> (<var>expr</var>, <var>x</var>)</i> |
|
550 |
<a name="IDX507"></a> |
|
551 |
</dt> |
|
552 |
<dt><u>Function:</u> <b>trigreduce</b><i> (<var>expr</var>)</i> |
|
553 |
<a name="IDX508"></a> |
|
554 |
</dt> |
|
555 |
<dd><p>Combines products and powers of trigonometric |
|
556 |
and hyperbolic sin's and cos's of <var>x</var> into those of multiples of <var>x</var>. |
|
557 |
It also tries to eliminate these functions when they occur in |
|
558 |
denominators. If <var>x</var> is omitted then all variables in <var>expr</var> are used. |
|
559 |
</p> |
|
560 |
<p>See also <code>poissimp</code>. |
|
561 |
</p> |
|
562 |
<table><tr><td> </td><td><pre class="example">(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x); |
|
563 |
cos(2 x) cos(2 x) 1 1 |
|
564 |
(%o1) -------- + 3 (-------- + -) + x - - |
|
565 |
2 2 2 2 |
|
566 |
||
567 |
</pre></td></tr></table> |
|
568 |
<p>The trigonometric simplification routines will use declared |
|
569 |
information in some simple cases. Declarations about variables are |
|
570 |
used as follows, e.g. |
|
571 |
</p> |
|
572 |
<table><tr><td> </td><td><pre class="example">(%i1) declare(j, integer, e, even, o, odd)$ |
|
573 |
(%i2) sin(x + (e + 1/2)*%pi); |
|
574 |
(%o2) cos(x) |
|
575 |
(%i3) sin(x + (o + 1/2)*%pi); |
|
576 |
(%o3) - cos(x) |
|
577 |
||
578 |
</pre></td></tr></table> |
|
579 |
</dd></dl> |
|
580 |
||
581 |
<dl> |
|
582 |
<dt><u>Option variable:</u> <b>trigsign</b> |
|
583 |
<a name="IDX509"></a> |
|
584 |
</dt> |
|
585 |
<dd><p>Default value: <code>true</code> |
|
586 |
</p> |
|
587 |
<p>When <code>trigsign</code> is <code>true</code>, it permits simplification of negative |
|
588 |
arguments to trigonometric functions. E.g., <code>sin(-x)</code> will become |
|
589 |
<code>-sin(x)</code> only if <code>trigsign</code> is <code>true</code>. |
|
590 |
</p> |
|
591 |
</dd></dl> |
|
592 |
||
593 |
<dl> |
|
594 |
<dt><u>Function:</u> <b>trigsimp</b><i> (<var>expr</var>)</i> |
|
595 |
<a name="IDX510"></a> |
|
596 |
</dt> |
|
597 |
<dd><p>Employs the identities <em>sin(x)^2 + cos(x)^2 = 1</em> and |
|
598 |
<em>cosh(x)^2 - sinh(x)^2 = 1</em> to simplify expressions containing <code>tan</code>, <code>sec</code>, |
|
599 |
etc., to <code>sin</code>, <code>cos</code>, <code>sinh</code>, <code>cosh</code>. |
|
600 |
</p> |
|
601 |
<p><code>trigreduce</code>, <code>ratsimp</code>, and <code>radcan</code> may be |
|
602 |
able to further simplify the result. |
|
603 |
</p> |
|
604 |
<p><code>demo ("trgsmp.dem")</code> displays some examples of <code>trigsimp</code>. |
|
605 |
</p> |
|
606 |
</dd></dl> |
|
607 |
||
608 |
<dl> |
|
609 |
<dt><u>Function:</u> <b>trigrat</b><i> (<var>expr</var>)</i> |
|
610 |
<a name="IDX511"></a> |
|
611 |
</dt> |
|
612 |
<dd><p>Gives a canonical simplifyed quasilinear form of a |
|
613 |
trigonometrical expression; <var>expr</var> is a rational fraction of several <code>sin</code>, |
|
614 |
<code>cos</code> or <code>tan</code>, the arguments of them are linear forms in some variables (or |
|
615 |
kernels) and <code>%pi/<var>n</var></code> (<var>n</var> integer) with integer coefficients. The result is a |
|
616 |
simplified fraction with numerator and denominator linear in <code>sin</code> and <code>cos</code>. |
|
617 |
Thus <code>trigrat</code> linearize always when it is possible. |
|
618 |
</p> |
|
619 |
<table><tr><td> </td><td><pre class="example">(%i1) trigrat(sin(3*a)/sin(a+%pi/3)); |
|
620 |
(%o1) sqrt(3) sin(2 a) + cos(2 a) - 1 |
|
621 |
||
622 |
</pre></td></tr></table> |
|
623 |
<p>The following example is taken from |
|
624 |
Davenport, Siret, and Tournier, <i>Calcul Formel</i>, Masson (or in English, |
|
625 |
Addison-Wesley), section 1.5.5, Morley theorem. |
|
626 |
</p> |
|
627 |
<table><tr><td> </td><td><pre class="example">(%i1) c: %pi/3 - a - b; |
|
628 |
%pi |
|
629 |
(%o1) - b - a + --- |
|
630 |
3 |
|
631 |
(%i2) bc: sin(a)*sin(3*c)/sin(a+b); |
|
632 |
sin(a) sin(3 b + 3 a) |
|
633 |
(%o2) --------------------- |
|
634 |
sin(b + a) |
|
635 |
(%i3) ba: bc, c=a, a=c$ |
|
636 |
(%i4) ac2: ba^2 + bc^2 - 2*bc*ba*cos(b); |
|
637 |
2 2 |
|
638 |
sin (a) sin (3 b + 3 a) |
|
639 |
(%o4) ----------------------- |
|
640 |
2 |
|
641 |
sin (b + a) |
|
642 |
||
643 |
%pi |
|
644 |
2 sin(a) sin(3 a) cos(b) sin(b + a - ---) sin(3 b + 3 a) |
|
645 |
3 |
|
646 |
- -------------------------------------------------------- |
|
647 |
%pi |
|
648 |
sin(a - ---) sin(b + a) |
|
649 |
3 |
|
650 |
||
651 |
2 2 %pi |
|
652 |
sin (3 a) sin (b + a - ---) |
|
653 |
3 |
|
654 |
+ --------------------------- |
|
655 |
2 %pi |
|
656 |
sin (a - ---) |
|
657 |
3 |
|
658 |
(%i5) trigrat (ac2); |
|
659 |
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a) |
|
660 |
||
661 |
- 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a) |
|
662 |
||
663 |
- 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a) |
|
664 |
||
665 |
+ 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a) |
|
666 |
||
667 |
+ sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b) |
|
668 |
||
669 |
+ sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a) |
|
670 |
||
671 |
- 9)/4 |
|
672 |
||
673 |
</pre></td></tr></table> |
|
674 |
</dd></dl> |
|
675 |
||
676 |
||
677 |
<hr size="6"> |
|
678 |
<table cellpadding="1" cellspacing="1" border="0"> |
|
679 |
<tr><td valign="middle" align="left">[<a href="#SEC51" title="Beginning of this chapter or previous chapter"> << </a>]</td> |
|
680 |
<td valign="middle" align="left">[<a href="maxima_16.html#SEC54" title="Next chapter"> >> </a>]</td> |
|
681 |
<td valign="middle" align="left"> </td> |
|
682 |
<td valign="middle" align="left"> </td> |
|
683 |
<td valign="middle" align="left"> </td> |
|
684 |
<td valign="middle" align="left"> </td> |
|
685 |
<td valign="middle" align="left"> </td> |
|
686 |
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td> |
|
687 |
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td> |
|
688 |
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td> |
|
689 |
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td> |
|
690 |
</tr></table> |
|
691 |
<p> |
|
692 |
<font size="-1"> |
|
693 |
This document was generated by <em>Robert Dodier</em> on <em>September, 20 2006</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>. |
|
694 |
</font> |
|
695 |
<br> |
|
696 |
||
697 |
</p> |
|
698 |
</body> |
|
699 |
</html> |