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<h1 class="chapter"> 15. Trigonometric </h1>
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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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<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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<a name="Introduction-to-Trigonometric"></a>
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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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<a name="Definitions-for-Trigonometric"></a>
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<h2 class="section"> 15.2 Definitions for Trigonometric </h2>
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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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<dd><p> - Arc Cosine.
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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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<dd><p> - Hyperbolic Arc Cosine.
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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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<dd><p> - Arc Cotangent.
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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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<dd><p> - Hyperbolic Arc Cotangent.
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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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<dd><p> - Arc Cosecant.
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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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<dd><p> - Hyperbolic Arc Cosecant.
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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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<dd><p> - Arc Secant.
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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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<dd><p> - Hyperbolic Arc Secant.
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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><u>Function:</u> <b>asinh</b><i> (<var>x</var>)</i>
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<a name="IDX484"></a>
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<dd><p> - Hyperbolic Arc Sine.
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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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<dd><p> - Arc Tangent.
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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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<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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<dt><u>Function:</u> <b>atanh</b><i> (<var>x</var>)</i>
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<a name="IDX487"></a>
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<dd><p> - Hyperbolic Arc Tangent.
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<dt><u>Package:</u> <b>atrig1</b>
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<a name="IDX488"></a>
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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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<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><u>Function:</u> <b>cosh</b><i> (<var>x</var>)</i>
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<a name="IDX490"></a>
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<dd><p> - Hyperbolic Cosine.
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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><u>Function:</u> <b>coth</b><i> (<var>x</var>)</i>
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<a name="IDX492"></a>
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<dd><p> - Hyperbolic Cotangent.
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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><u>Function:</u> <b>csch</b><i> (<var>x</var>)</i>
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<a name="IDX494"></a>
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<dd><p> - Hyperbolic Cosecant.
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<dt><u>Option variable:</u> <b>halfangles</b>
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<a name="IDX495"></a>
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<dd><p>Default value: <code>false</code>
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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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<dt><u>Package:</u> <b>ntrig</b>
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<a name="IDX496"></a>
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<dd><p>The <code>ntrig</code> package contains a set of simplification rules that are
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used to simplify trigonometric function whose arguments are of the form
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<code><var>f</var>(<var>n</var> %pi/10)</code> where <var>f</var> is any of the functions
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<code>sin</code>, <code>cos</code>, <code>tan</code>, <code>csc</code>, <code>sec</code> and <code>cot</code>.
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<dt><u>Function:</u> <b>sec</b><i> (<var>x</var>)</i>
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<a name="IDX497"></a>
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<dt><u>Function:</u> <b>sech</b><i> (<var>x</var>)</i>
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<a name="IDX498"></a>
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<dd><p> - Hyperbolic Secant.
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<dt><u>Function:</u> <b>sin</b><i> (<var>x</var>)</i>
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<a name="IDX499"></a>
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<dt><u>Function:</u> <b>sinh</b><i> (<var>x</var>)</i>
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<a name="IDX500"></a>
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<dd><p> - Hyperbolic Sine.
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<dt><u>Function:</u> <b>tan</b><i> (<var>x</var>)</i>
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<a name="IDX501"></a>
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<dt><u>Function:</u> <b>tanh</b><i> (<var>x</var>)</i>
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<a name="IDX502"></a>
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<dd><p> - Hyperbolic Tangent.
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<dt><u>Function:</u> <b>trigexpand</b><i> (<var>expr</var>)</i>
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<a name="IDX503"></a>
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<dd><p>Expands trigonometric and hyperbolic functions of
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sums of angles and of multiple angles occurring in <var>expr</var>. For best
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results, <var>expr</var> should be expanded. To enhance user control of
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simplification, this function expands only one level at a time,
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expanding sums of angles or multiple angles. To obtain full expansion
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into sines and cosines immediately, set the switch <code>trigexpand: true</code>.
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<p><code>trigexpand</code> is governed by the following global flags:
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<dl compact="compact">
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<dt> <code>trigexpand</code></dt>
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<dd><p>If <code>true</code> causes expansion of all
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expressions containing sin's and cos's occurring subsequently.
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<dt> <code>halfangles</code></dt>
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<dd><p>If <code>true</code> causes half-angles to be simplified
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<dt> <code>trigexpandplus</code></dt>
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<dd><p>Controls the "sum" rule for <code>trigexpand</code>,
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expansion of sums (e.g. <code>sin(x + y)</code>) will take place only if
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<code>trigexpandplus</code> is <code>true</code>.
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<dt> <code>trigexpandtimes</code></dt>
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<dd><p>Controls the "product" rule for <code>trigexpand</code>,
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expansion of products (e.g. <code>sin(2 x)</code>) will take place only if
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<code>trigexpandtimes</code> is <code>true</code>.
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<table><tr><td> </td><td><pre class="example">(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
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(%o1) - sin (x) + 3 cos (x) + x
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(%i2) trigexpand(sin(10*x+y));
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(%o2) cos(10 x) sin(y) + sin(10 x) cos(y)
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</pre></td></tr></table>
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<dt><u>Option variable:</u> <b>trigexpandplus</b>
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<a name="IDX504"></a>
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<dd><p>Default value: <code>true</code>
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<p><code>trigexpandplus</code> controls the "sum" rule for
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<code>trigexpand</code>. Thus, when the <code>trigexpand</code> command is used or the
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<code>trigexpand</code> switch set to <code>true</code>, expansion of sums
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(e.g. <code>sin(x+y))</code> will take place only if <code>trigexpandplus</code> is
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<dt><u>Option variable:</u> <b>trigexpandtimes</b>
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<a name="IDX505"></a>
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<dd><p>Default value: <code>true</code>
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<p><code>trigexpandtimes</code> controls the "product" rule for
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<code>trigexpand</code>. Thus, when the <code>trigexpand</code> command is used or the
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<code>trigexpand</code> switch set to <code>true</code>, expansion of products (e.g. <code>sin(2*x)</code>)
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will take place only if <code>trigexpandtimes</code> is <code>true</code>.
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<dt><u>Option variable:</u> <b>triginverses</b>
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<a name="IDX506"></a>
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<dd><p>Default value: <code>all</code>
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<p><code>triginverses</code> controls the simplification of the
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composition of trigonometric and hyperbolic functions with their inverse
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<p>If <code>all</code>, both e.g. <code>atan(tan(<var>x</var>))</code>
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and <code>tan(atan(<var>x</var>))</code> simplify to <var>x</var>.
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<p>If <code>true</code>, the <code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code>
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simplification is turned off.
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<p>If <code>false</code>, both the
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<code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code> and
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<code><var>fun</var>(<var>arcfun</var>(<var>x</var>))</code>
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simplifications are turned off.
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<dt><u>Function:</u> <b>trigreduce</b><i> (<var>expr</var>, <var>x</var>)</i>
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<a name="IDX507"></a>
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<dt><u>Function:</u> <b>trigreduce</b><i> (<var>expr</var>)</i>
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<a name="IDX508"></a>
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<dd><p>Combines products and powers of trigonometric
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and hyperbolic sin's and cos's of <var>x</var> into those of multiples of <var>x</var>.
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It also tries to eliminate these functions when they occur in
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denominators. If <var>x</var> is omitted then all variables in <var>expr</var> are used.
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<p>See also <code>poissimp</code>.
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<table><tr><td> </td><td><pre class="example">(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
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cos(2 x) cos(2 x) 1 1
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(%o1) -------- + 3 (-------- + -) + x - -
567
</pre></td></tr></table>
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<p>The trigonometric simplification routines will use declared
569
information in some simple cases. Declarations about variables are
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used as follows, e.g.
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<table><tr><td> </td><td><pre class="example">(%i1) declare(j, integer, e, even, o, odd)$
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(%i2) sin(x + (e + 1/2)*%pi);
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(%i3) sin(x + (o + 1/2)*%pi);
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</pre></td></tr></table>
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<dt><u>Option variable:</u> <b>trigsign</b>
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<a name="IDX509"></a>
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<dd><p>Default value: <code>true</code>
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<p>When <code>trigsign</code> is <code>true</code>, it permits simplification of negative
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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>.
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<dt><u>Function:</u> <b>trigsimp</b><i> (<var>expr</var>)</i>
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<a name="IDX510"></a>
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<dd><p>Employs the identities <em>sin(x)^2 + cos(x)^2 = 1</em> and
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<em>cosh(x)^2 - sinh(x)^2 = 1</em> to simplify expressions containing <code>tan</code>, <code>sec</code>,
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etc., to <code>sin</code>, <code>cos</code>, <code>sinh</code>, <code>cosh</code>.
601
<p><code>trigreduce</code>, <code>ratsimp</code>, and <code>radcan</code> may be
602
able to further simplify the result.
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<p><code>demo ("trgsmp.dem")</code> displays some examples of <code>trigsimp</code>.
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<dt><u>Function:</u> <b>trigrat</b><i> (<var>expr</var>)</i>
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<a name="IDX511"></a>
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<dd><p>Gives a canonical simplifyed quasilinear form of a
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trigonometrical expression; <var>expr</var> is a rational fraction of several <code>sin</code>,
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<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.
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
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.
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<table><tr><td> </td><td><pre class="example">(%i1) c: %pi/3 - a - b;
631
(%i2) bc: sin(a)*sin(3*c)/sin(a+b);
632
sin(a) sin(3 b + 3 a)
633
(%o2) ---------------------
635
(%i3) ba: bc, c=a, a=c$
636
(%i4) ac2: ba^2 + bc^2 - 2*bc*ba*cos(b);
638
sin (a) sin (3 b + 3 a)
639
(%o4) -----------------------
644
2 sin(a) sin(3 a) cos(b) sin(b + a - ---) sin(3 b + 3 a)
646
- --------------------------------------------------------
648
sin(a - ---) sin(b + a)
652
sin (3 a) sin (b + a - ---)
654
+ ---------------------------
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(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
661
- 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
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- 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
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+ 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
667
+ sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
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+ sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
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</pre></td></tr></table>
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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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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>.
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