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Next: <a href="maxima_58.html#Additional-Functions" accesskey="n" rel="next">Additional Functions</a>, Previous: <a href="maxima_56.html#Options-Controlling-Simplification" accesskey="p" rel="previous">Options Controlling Simplification</a>, Up: <a href="maxima_54.html#Functions-and-Variables-for-Trigonometric" accesskey="u" rel="up">Functions and Variables for Trigonometric</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Explicit-Simplifications-Using-Identities-1"></a>
<h4 class="subsubsection">10.5.2.3 Explicit Simplifications Using Identities</h4>
<a name="trigexpand"></a><a name="Item_003a-MathFunctions_002fdeffn_002ftrigexpand"></a><dl>
<dt><a name="index-trigexpand"></a>Function: <strong>trigexpand</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Expands trigonometric and hyperbolic functions of
sums of angles and of multiple angles occurring in <var>expr</var>. For best
results, <var>expr</var> should be expanded. To enhance user control of
simplification, this function expands only one level at a time,
expanding sums of angles or multiple angles. To obtain full expansion
into sines and cosines immediately, set the switch <code>trigexpand: true</code>.
</p>
<p><code>trigexpand</code> is governed by the following global flags:
</p>
<dl compact="compact">
<dt><code><a href="#trigexpand">trigexpand</a></code></dt>
<dd><p>If <code>true</code> causes expansion of all
expressions containing sin’s and cos’s occurring subsequently.
</p></dd>
<dt><code><a href="maxima_56.html#halfangles">halfangles</a></code></dt>
<dd><p>If <code>true</code> causes half-angles to be simplified
away.
</p></dd>
<dt><code><a href="#trigexpandplus">trigexpandplus</a></code></dt>
<dd><p>Controls the "sum" rule for <code>trigexpand</code>,
expansion of sums (e.g. <code>sin(x + y)</code>) will take place only if
<code>trigexpandplus</code> is <code>true</code>.
</p></dd>
<dt><code><a href="#trigexpandtimes">trigexpandtimes</a></code></dt>
<dd><p>Controls the "product" rule for <code>trigexpand</code>,
expansion of products (e.g. <code>sin(2 x)</code>) will take place only if
<code>trigexpandtimes</code> is <code>true</code>.
</p></dd>
</dl>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
2 2
(%o1) (- sin (x)) + 3 cos (x) + x
</pre><pre class="example">(%i2) trigexpand(sin(10*x+y));
(%o2) cos(10 x) sin(y) + sin(10 x) cos(y)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div></dd></dl>
<a name="trigexpandplus"></a><a name="Item_003a-MathFunctions_002fdefvr_002ftrigexpandplus"></a><dl>
<dt><a name="index-trigexpandplus"></a>Option variable: <strong>trigexpandplus</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>trigexpandplus</code> controls the "sum" rule for
<code><a href="#trigexpand">trigexpand</a></code>. Thus, when the <code><a href="#trigexpand">trigexpand</a></code> command is used or the
<code><a href="#trigexpand">trigexpand</a></code> switch set to <code>true</code>, expansion of sums
(e.g. <code>sin(x+y))</code> will take place only if <code>trigexpandplus</code> is
<code>true</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="trigexpandtimes"></a><a name="Item_003a-MathFunctions_002fdefvr_002ftrigexpandtimes"></a><dl>
<dt><a name="index-trigexpandtimes"></a>Option variable: <strong>trigexpandtimes</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>trigexpandtimes</code> controls the "product" rule for <code><a href="#trigexpand">trigexpand</a></code>.
Thus, when the <code><a href="#trigexpand">trigexpand</a></code> command is used or the <code><a href="#trigexpand">trigexpand</a></code>
switch set to <code>true</code>, expansion of products (e.g. <code>sin(2*x)</code>)
will take place only if <code>trigexpandtimes</code> is <code>true</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="triginverses"></a><a name="Item_003a-MathFunctions_002fdefvr_002ftriginverses"></a><dl>
<dt><a name="index-triginverses"></a>Option variable: <strong>triginverses</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>triginverses</code> controls the simplification of the
composition of trigonometric and hyperbolic functions with their inverse
functions.
</p>
<p>If <code>all</code>, both e.g. <code>atan(tan(<var>x</var>))</code>
and <code>tan(atan(<var>x</var>))</code> simplify to <var>x</var>.
</p>
<p>If <code>true</code>, the <code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code>
simplification is turned off.
</p>
<p>If <code>false</code>, both the
<code><var>arcfun</var>(<var>fun</var>(<var>x</var>))</code> and
<code><var>fun</var>(<var>arcfun</var>(<var>x</var>))</code>
simplifications are turned off.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="trigreduce"></a><a name="Item_003a-MathFunctions_002fdeffn_002ftrigreduce"></a><dl>
<dt><a name="index-trigreduce"></a>Function: <strong>trigreduce</strong> <em><br> <tt>trigreduce</tt> (<var>expr</var>, <var>x</var>) <br> <tt>trigreduce</tt> (<var>expr</var>)</em></dt>
<dd>
<p>Combines products and powers of trigonometric
and hyperbolic sin’s and cos’s of <var>x</var> into those of multiples of <var>x</var>.
It also tries to eliminate these functions when they occur in
denominators. If <var>x</var> is omitted then all variables in <var>expr</var> are used.
</p>
<p>See also <code><a href="maxima_144.html#poissimp">poissimp</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
cos(2 x) cos(2 x) 1 1
(%o1) -------- + 3 (-------- + -) + x - -
2 2 2 2
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div></dd></dl>
<a name="trigsimp"></a><a name="Item_003a-MathFunctions_002fdeffn_002ftrigsimp"></a><dl>
<dt><a name="index-trigsimp"></a>Function: <strong>trigsimp</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Employs the identities
\(\sin\left(x\right)^2 + \cos\left(x\right)^2 = 1\) and
\(\cosh\left(x\right)^2 - \sinh\left(x\right)^2 = 1\) to
simplify expressions containing <code>tan</code>, <code>sec</code>,
etc., to <code>sin</code>, <code>cos</code>, <code>sinh</code>, <code>cosh</code>.
</p>
<p><code><a href="#trigreduce">trigreduce</a></code>, <code><a href="maxima_80.html#ratsimp">ratsimp</a></code>, and <code><a href="maxima_46.html#radcan">radcan</a></code> may be
able to further simplify the result.
</p>
<p><code>demo ("trgsmp.dem")</code> displays some examples of <code>trigsimp</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div></dd></dl>
<a name="trigrat"></a><a name="Item_003a-MathFunctions_002fdeffn_002ftrigrat"></a><dl>
<dt><a name="index-trigrat"></a>Function: <strong>trigrat</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Gives a canonical simplified quasilinear form of a trigonometrical expression;
<var>expr</var> is a rational fraction of several <code>sin</code>, <code>cos</code> or
<code>tan</code>, the arguments of them are linear forms in some variables (or
kernels) and <code>%pi/<var>n</var></code> (<var>n</var> integer) with integer coefficients.
The result is a simplified fraction with numerator and denominator linear in
<code>sin</code> and <code>cos</code>. Thus <code>trigrat</code> linearize always when it is
possible.
</p>
<div class="example">
<pre class="example">(%i1) trigrat(sin(3*a)/sin(a+%pi/3));
(%o1) sqrt(3) sin(2 a) + cos(2 a) - 1
</pre></div>
<p>The following example is taken from
Davenport, Siret, and Tournier, <i>Calcul Formel</i>, Masson (or in English,
Addison-Wesley), section 1.5.5, Morley theorem.
</p>
<div class="example">
<pre class="example">(%i1) c : %pi/3 - a - b$
</pre><pre class="example">(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
%pi
sin(a) sin(3 ((- b) - a + ---))
3
(%o2) -------------------------------
sin(b + a)
</pre><pre class="example">(%i3) ba : bc, c=a, a=c;
%pi
sin(3 a) sin(b + a - ---)
3
(%o3) -------------------------
%pi
sin(a - ---)
3
</pre><pre class="example">(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
2 2 %pi
sin (3 a) sin (b + a - ---)
3
(%o4) ---------------------------
2 %pi
sin (a - ---)
3
%pi
- (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
3
%pi %pi
sin(b + a - ---))/(sin(a - ---) sin(b + a))
3 3
2 2 %pi
sin (a) sin (3 ((- b) - a + ---))
3
+ ---------------------------------
2
sin (b + a)
</pre><pre class="example">(%i5) trigrat (ac2);
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
- 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
- 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
+ 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
+ sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
+ sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
- 9)/4
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Trigonometric-functions">Trigonometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·</div></dd></dl>
<a name="Item_003a-MathFunctions_002fnode_002fAdditional-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_58.html#Additional-Functions" accesskey="n" rel="next">Additional Functions</a>, Previous: <a href="maxima_56.html#Options-Controlling-Simplification" accesskey="p" rel="previous">Options Controlling Simplification</a>, Up: <a href="maxima_54.html#Functions-and-Variables-for-Trigonometric" accesskey="u" rel="up">Functions and Variables for Trigonometric</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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