1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima 5.47.0 Manual: Options Controlling Simplification</title>
<meta name="description" content="Maxima 5.47.0 Manual: Options Controlling Simplification">
<meta name="keywords" content="Maxima 5.47.0 Manual: Options Controlling Simplification">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link href="maxima_toc.html#Top" rel="start" title="Top">
<link href="maxima_423.html#Function-and-Variable-Index" rel="index" title="Function and Variable Index">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_54.html#Functions-and-Variables-for-Trigonometric" rel="up" title="Functions and Variables for Trigonometric">
<link href="maxima_57.html#Explicit-Simplifications-Using-Identities" rel="next" title="Explicit Simplifications Using Identities">
<link href="maxima_55.html#Trigonometric-and-Hyperbolic-Functions" rel="previous" title="Trigonometric and Hyperbolic Functions">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
body {color: black; background: white; margin-left: 8%; margin-right: 13%;
font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
font-family: sans-serif}
div.synopsisbox {
border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 1em; padding-bottom: 1em;
padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}
-->
</style>
<link rel="icon" href="figures/favicon.ico">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Options-Controlling-Simplification"></a>
<div class="header">
<p>
Next: <a href="maxima_57.html#Explicit-Simplifications-Using-Identities" accesskey="n" rel="next">Explicit Simplifications Using Identities</a>, Previous: <a href="maxima_55.html#Trigonometric-and-Hyperbolic-Functions" accesskey="p" rel="previous">Trigonometric and Hyperbolic Functions</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="Options-Controlling-Simplification-1"></a>
<h4 class="subsubsection">10.5.2.2 Options Controlling Simplification</h4>
<a name="g_t_0025piargs"></a><a name="Item_003a-MathFunctions_002fdefvr_002f_0025piargs"></a><dl>
<dt><a name="index-_0025piargs"></a>Option variable: <strong>%piargs</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>%piargs</code> is <code>true</code>,
trigonometric functions are simplified to algebraic constants
when the argument is an integer multiple
of
\(\pi,\)<!-- /@w -->
\(\pi/2,\)<!-- /@w -->
\(\pi/4,\)<!-- /@w --> or
\(\pi/6.\)</p>
<p>Maxima knows some identities which can be applied when
\(\pi\), etc.,
are multiplied by an integer variable (that is, a symbol declared to be
integer).
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) %piargs : false$
</pre><pre class="example">(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
%pi %pi
(%o2) [sin(%pi), sin(---), sin(---)]
2 3
</pre><pre class="example">(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
%pi %pi %pi
(%o3) [sin(---), sin(---), sin(---)]
4 5 6
</pre><pre class="example">(%i4) %piargs : true$
</pre><pre class="example">(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
sqrt(3)
(%o5) [0, 1, -------]
2
</pre><pre class="example">(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
1 %pi 1
(%o6) [-------, sin(---), -]
sqrt(2) 5 2
</pre><pre class="example">(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
cos (sqrt(2)*%pi/3)];
1 1 sqrt(2) %pi
(%o7) [-, - -, sqrt(3), cos(-----------)]
2 2 3
</pre></div>
<p>Some identities are applied when
\(\pi\) and
\(\pi/2\) are
multiplied by an integer variable.
</p>
<div class="example">
<pre class="example">(%i1) declare (n, integer, m, even)$
</pre><pre class="example">(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
cos (%pi/2 * m)];
m/2
(%o2) [0, 1, 0, (- 1) ]
</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-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="g_t_0025iargs"></a><a name="Item_003a-MathFunctions_002fdefvr_002f_0025iargs"></a><dl>
<dt><a name="index-_0025iargs"></a>Option variable: <strong>%iargs</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>%iargs</code> is <code>true</code>,
trigonometric functions are simplified to hyperbolic functions
when the argument is apparently a multiple of the imaginary
unit
\(i.\)</p>
<p>Even when the argument is demonstrably real, the simplification is applied;
Maxima considers only whether the argument is a literal multiple
of
\(i.\)</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) %iargs : false$
</pre><pre class="example">(%i2) [sin (%i * x), cos (%i * x), tan (%i * x)];
(%o2) [sin(%i x), cos(%i x), tan(%i x)]
</pre><pre class="example">(%i3) %iargs : true$
</pre><pre class="example">(%i4) [sin (%i * x), cos (%i * x), tan (%i * x)];
(%o4) [%i sinh(x), cosh(x), %i tanh(x)]
</pre></div>
<p>Even when the argument is demonstrably real, the simplification is applied.
</p>
<div class="example">
<pre class="example">(%i1) declare (x, imaginary)$
</pre><pre class="example">(%i2) [featurep (x, imaginary), featurep (x, real)];
(%o2) [true, false]
</pre><pre class="example">(%i3) sin (%i * x);
(%o3) %i sinh(x)
</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-Hyperbolic-functions">Hyperbolic functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="halfangles"></a><a name="Item_003a-MathFunctions_002fdefvr_002fhalfangles"></a><dl>
<dt><a name="index-halfangles"></a>Option variable: <strong>halfangles</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>halfangles</code> is <code>true</code>, trigonometric functions of arguments
<code><var>expr</var>/2</code> are simplified to functions of <var>expr</var>.
</p>
<p>For a real argument <em>x</em> in the interval
<!-- /@w -->
\(0 \le x < 2\pi,\)<!-- /@w -->
\(\sin{x\over 2}\) simplifies to a simple formula:
$$
{\sqrt{1-\cos x}\over\sqrt{2}}
$$</p>
<p>A complicated factor is needed to make this formula correct for all complex
arguments <em>z = x+iy</em>:
$$
(-1)^{\lfloor{x/(2\pi)}\rfloor}
\left[1-\rm{unit\_step}(-y)
\left(1+(-1)^{\lfloor{x/(2\pi)}\rfloor - \lceil{x/(2\pi)}\rceil}\right)\right]
$$</p>
<p>Maxima knows this factor and similar factors for the functions <code>sin</code>,
<code>cos</code>, <code>sinh</code>, and <code>cosh</code>. For special values of the argument
<em>z</em> these factors simplify accordingly.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) halfangles : false$
</pre><pre class="example">(%i2) sin (x / 2);
x
(%o2) sin(-)
2
</pre><pre class="example">(%i3) halfangles : true$
</pre><pre class="example">(%i4) sin (x / 2);
x
floor(-----)
2 %pi
(- 1) sqrt(1 - cos(x))
(%o4) ----------------------------------
sqrt(2)
</pre><pre class="example">(%i5) assume(x>0, x<2*%pi)$
</pre><pre class="example">(%i6) sin(x / 2);
sqrt(1 - cos(x))
(%o6) ----------------
sqrt(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-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="trigsign"></a><a name="Item_003a-MathFunctions_002fdefvr_002ftrigsign"></a><dl>
<dt><a name="index-trigsign"></a>Option variable: <strong>trigsign</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>trigsign</code> is <code>true</code>, it permits simplification of negative
arguments to trigonometric functions. E.g.,
\(\sin(-x)\) will
become
\(-\sin x\) only if <code>trigsign</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="Item_003a-MathFunctions_002fnode_002fExplicit-Simplifications-Using-Identities"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_57.html#Explicit-Simplifications-Using-Identities" accesskey="n" rel="next">Explicit Simplifications Using Identities</a>, Previous: <a href="maxima_55.html#Trigonometric-and-Hyperbolic-Functions" accesskey="p" rel="previous">Trigonometric and Hyperbolic Functions</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>
</body>
</html>
|
