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<div class="section" id="trigonometric-functions">
<h1>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h1>
<p>Except where otherwise noted, the trigonometric functions
take a radian angle as input and the inverse trigonometric
functions return radian angles.</p>
<p>The ordinary trigonometric functions are single-valued
functions defined everywhere in the complex plane
(except at the poles of tan, sec, csc, and cot).
They are defined generally via the exponential function,
e.g.</p>
<div class="math">
<p><img src="../_images/math/0086a58c3ae9fd951eadf3111b11b80b19366d0d.png" alt="\cos(x) = \frac{e^{ix} + e^{-ix}}{2}." /></p>
</div><p>The inverse trigonometric functions are multivalued,
thus requiring branch cuts, and are generally real-valued
only on a part of the real line. Definitions and branch cuts
are given in the documentation of each function.
The branch cut conventions used by mpmath are essentially
the same as those found in most standard mathematical software,
such as Mathematica and Python’s own <tt class="docutils literal"><span class="pre">cmath</span></tt> libary (as of Python 2.6;
earlier Python versions implement some functions
erroneously).</p>
<div class="section" id="degree-radian-conversion">
<h2>Degree-radian conversion<a class="headerlink" href="#degree-radian-conversion" title="Permalink to this headline">¶</a></h2>
<div class="section" id="degrees">
<h3><tt class="xref docutils literal"><span class="pre">degrees()</span></tt><a class="headerlink" href="#degrees" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.degrees">
<tt class="descclassname">mpmath.</tt><tt class="descname">degrees</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.degrees" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts the radian angle <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> to a degree angle:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">degrees</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">60.0</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="radians">
<h3><tt class="xref docutils literal"><span class="pre">radians()</span></tt><a class="headerlink" href="#radians" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.radians">
<tt class="descclassname">mpmath.</tt><tt class="descname">radians</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.radians" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts the degree angle <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> to radians:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">radians</span><span class="p">(</span><span class="mi">60</span><span class="p">)</span>
<span class="go">1.0471975511966</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="id1">
<h2>Trigonometric functions<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
<div class="section" id="cos">
<h3><tt class="xref docutils literal"><span class="pre">cos()</span></tt><a class="headerlink" href="#cos" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.cos">
<tt class="descclassname">mpmath.</tt><tt class="descname">cos</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cos" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the cosine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/c938e3f9dea789595c6290070f7975f45bc4300a.png" alt="\cos(x)"/>.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">cos</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
<span class="go">-0.9802850113244713353133243</span>
<span class="gp">>>> </span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-4.189625690968807230132555 - 9.109227893755336597979197j)</span>
<span class="gp">>>> </span><span class="n">cos</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[1.0, 0.0, -0.5, 0.0, 0.0416667, 0.0, -0.00138889]</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.cos()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">cos</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[0.540302305868139717400936602301, 1.0]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">cos</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="go">[-0.41614683654714238699756823214, 1.0]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sin">
<h3><tt class="xref docutils literal"><span class="pre">sin()</span></tt><a class="headerlink" href="#sin" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sin">
<tt class="descclassname">mpmath.</tt><tt class="descname">sin</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sin" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the sine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/081bf0417da2140657aad909a14ca6c3ff8b77d5.png" alt="\sin(x)"/>.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.8660254037844386467637232</span>
<span class="gp">>>> </span><span class="n">sin</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
<span class="go">0.1975887055794968911438743</span>
<span class="gp">>>> </span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(9.1544991469114295734673 - 4.168906959966564350754813j)</span>
<span class="gp">>>> </span><span class="n">sin</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333, 0.0]</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.sin()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">sin</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[0.0, 0.841470984807896506652502331201]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">sin</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="go">[0.0, 1.0]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="tan">
<h3><tt class="xref docutils literal"><span class="pre">tan()</span></tt><a class="headerlink" href="#tan" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.tan">
<tt class="descclassname">mpmath.</tt><tt class="descname">tan</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.tan" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the tangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/cde42c84b80794d947bff0115c68e9d9992d39fb.png" alt="\tan(x) = \frac{\sin(x)}{\cos(x)}"/>.
The tangent function is singular at <img class="math" src="../_images/math/6fc799cb560279e3e2359ca1ae33ac57867265ce.png" alt="x = (n+1/2)\pi"/>, but
<tt class="docutils literal"><span class="pre">tan(x)</span></tt> always returns a finite result since <img class="math" src="../_images/math/b0055c4425e492c37e317d03a83dcf382f1da9ab.png" alt="(n+1/2)\pi"/>
cannot be represented exactly using floating-point arithmetic.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">tan</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">1.732050807568877293527446</span>
<span class="gp">>>> </span><span class="n">tan</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
<span class="go">-0.2015625081449864533091058</span>
<span class="gp">>>> </span><span class="n">tan</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-0.003764025641504248292751221 + 1.003238627353609801446359j)</span>
<span class="gp">>>> </span><span class="n">tan</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">tan</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[0.0, 1.0, 0.0, 0.333333, 0.0, 0.133333, 0.0]</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.tan()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">tan</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[0.0, 1.55740772465490223050697482944]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">tan</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="c"># Interval includes a singularity</span>
<span class="go">[-inf, +inf]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sec">
<h3><tt class="xref docutils literal"><span class="pre">sec()</span></tt><a class="headerlink" href="#sec" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sec">
<tt class="descclassname">mpmath.</tt><tt class="descname">sec</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sec" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the secant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/0875f33b2dab3f8884ffd45ec7c50d2c50825d17.png" alt="\mathrm{sec}(x) = \frac{1}{\cos(x)}"/>.
The secant function is singular at <img class="math" src="../_images/math/6fc799cb560279e3e2359ca1ae33ac57867265ce.png" alt="x = (n+1/2)\pi"/>, but
<tt class="docutils literal"><span class="pre">sec(x)</span></tt> always returns a finite result since <img class="math" src="../_images/math/b0055c4425e492c37e317d03a83dcf382f1da9ab.png" alt="(n+1/2)\pi"/>
cannot be represented exactly using floating-point arithmetic.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">sec</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">2.0</span>
<span class="gp">>>> </span><span class="n">sec</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
<span class="go">-1.184723164360392819100265</span>
<span class="gp">>>> </span><span class="n">sec</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-0.04167496441114427004834991 + 0.0906111371962375965296612j)</span>
<span class="gp">>>> </span><span class="n">sec</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sec</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[1.0, 0.0, 0.5, 0.0, 0.208333, 0.0, 0.0847222]</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.sec()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">sec</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[1.0, 1.85081571768092561791175326276]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">sec</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="c"># Interval includes a singularity</span>
<span class="go">[-inf, +inf]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="csc">
<h3><tt class="xref docutils literal"><span class="pre">csc()</span></tt><a class="headerlink" href="#csc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.csc">
<tt class="descclassname">mpmath.</tt><tt class="descname">csc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.csc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the cosecant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/734e0b59c6461bcef90c1e23339890fc4a078d97.png" alt="\mathrm{csc}(x) = \frac{1}{\sin(x)}"/>.
This cosecant function is singular at <img class="math" src="../_images/math/8742afe12de27fa28d4092f276aacff913ad9935.png" alt="x = n \pi"/>, but with the
exception of the point <img class="math" src="../_images/math/2d348bde3e15456e71734dc2c56fc7425c95927f.png" alt="x = 0"/>, <tt class="docutils literal"><span class="pre">csc(x)</span></tt> returns a finite result
since <img class="math" src="../_images/math/34efe5253c7b569cf0ec1641dfe9a2c4bdd367b6.png" alt="n \pi"/> cannot be represented exactly using floating-point
arithmetic.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">csc</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">1.154700538379251529018298</span>
<span class="gp">>>> </span><span class="n">csc</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
<span class="go">-1.864910497503629858938891</span>
<span class="gp">>>> </span><span class="n">csc</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.09047320975320743980579048 + 0.04120098628857412646300981j)</span>
<span class="gp">>>> </span><span class="n">csc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.csc()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">csc</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span> <span class="c"># Interval includes a singularity</span>
<span class="go">[1.18839510577812121626159943988, +inf]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">csc</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="go">[1.0, +inf]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="cot">
<h3><tt class="xref docutils literal"><span class="pre">cot()</span></tt><a class="headerlink" href="#cot" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.cot">
<tt class="descclassname">mpmath.</tt><tt class="descname">cot</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.cot" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the cotangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/4f2c1c156107fb3b33a9925ca78e282ac377f574.png" alt="\mathrm{cot}(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}"/>.
This cotangent function is singular at <img class="math" src="../_images/math/8742afe12de27fa28d4092f276aacff913ad9935.png" alt="x = n \pi"/>, but with the
exception of the point <img class="math" src="../_images/math/2d348bde3e15456e71734dc2c56fc7425c95927f.png" alt="x = 0"/>, <tt class="docutils literal"><span class="pre">cot(x)</span></tt> returns a finite result
since <img class="math" src="../_images/math/34efe5253c7b569cf0ec1641dfe9a2c4bdd367b6.png" alt="n \pi"/> cannot be represented exactly using floating-point
arithmetic.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">cot</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.5773502691896257645091488</span>
<span class="gp">>>> </span><span class="n">cot</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
<span class="go">1.574131876209625656003562</span>
<span class="gp">>>> </span><span class="n">cot</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">(-0.003739710376336956660117409 - 0.9967577965693583104609688j)</span>
<span class="gp">>>> </span><span class="n">cot</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">nan</span>
</pre></div>
</div>
<p>Intervals are supported via <tt class="xref docutils literal"><span class="pre">mpmath.iv.cot()</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">cot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span> <span class="c"># Interval includes a singularity</span>
<span class="go">[0.642092615934330703006419974862, +inf]</span>
<span class="gp">>>> </span><span class="n">iv</span><span class="o">.</span><span class="n">cot</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="go">[-inf, +inf]</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="trigonometric-functions-with-modified-argument">
<h2>Trigonometric functions with modified argument<a class="headerlink" href="#trigonometric-functions-with-modified-argument" title="Permalink to this headline">¶</a></h2>
<div class="section" id="cospi">
<h3><tt class="xref docutils literal"><span class="pre">cospi()</span></tt><a class="headerlink" href="#cospi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.cospi">
<tt class="descclassname">mpmath.</tt><tt class="descname">cospi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cospi" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes <img class="math" src="../_images/math/d49c1e556b2fd2cb785ca3fe0d14d1247707b79a.png" alt="\cos(\pi x)"/>, more accurately than the expression
<tt class="docutils literal"><span class="pre">cos(pi*x)</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">cospi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">))</span>
<span class="go">(1.0, 0.999999999997493)</span>
<span class="gp">>>> </span><span class="n">cospi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mf">0.5</span><span class="p">))</span>
<span class="go">(0.0, 1.59960492420134e-6)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sinpi">
<h3><tt class="xref docutils literal"><span class="pre">sinpi()</span></tt><a class="headerlink" href="#sinpi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sinpi">
<tt class="descclassname">mpmath.</tt><tt class="descname">sinpi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sinpi" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes <img class="math" src="../_images/math/61395a5902d5ef1f78b3dc6e004e7856f9e45c71.png" alt="\sin(\pi x)"/>, more accurately than the expression
<tt class="docutils literal"><span class="pre">sin(pi*x)</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">sinpi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">))</span>
<span class="go">(0.0, -2.23936276195592e-6)</span>
<span class="gp">>>> </span><span class="n">sinpi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mf">0.5</span><span class="p">))</span>
<span class="go">(1.0, 0.999999999998721)</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="inverse-trigonometric-functions">
<h2>Inverse trigonometric functions<a class="headerlink" href="#inverse-trigonometric-functions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="acos">
<h3><tt class="xref docutils literal"><span class="pre">acos()</span></tt><a class="headerlink" href="#acos" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acos">
<tt class="descclassname">mpmath.</tt><tt class="descname">acos</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.acos" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse cosine or arccosine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/42047524eb9efe8ffc20cac3c0f5291ad27e379c.png" alt="\cos^{-1}(x)"/>.
Since <img class="math" src="../_images/math/fc4162d019d241cabb62fd2c1d3a02213b8a9301.png" alt="-1 \le \cos(x) \le 1"/> for real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, the inverse
cosine is real-valued only for <img class="math" src="../_images/math/f260c78fb4c935f83297e9c2cb2cf7c3c3e3adfc.png" alt="-1 \le x \le 1"/>. On this interval,
<a title="mpmath.acos" class="reference internal" href="#mpmath.acos"><tt class="xref docutils literal"><span class="pre">acos()</span></tt></a> is defined to be a monotonically decreasing
function assuming values between <img class="math" src="../_images/math/08e02a30b63dba4891fe31161e3c4a2dc4dc4b2e.png" alt="+\pi"/> and <img class="math" src="../_images/math/bc1f9d9bf8a1b606a4188b5ce9a2af1809e27a89.png" alt="0"/>.</p>
<p>Basic values are:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">acos</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">3.141592653589793238462643</span>
<span class="gp">>>> </span><span class="n">acos</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">acos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">acos</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[1.5708, -1.0, 0.0, -0.166667, 0.0, -0.075, 0.0]</span>
</pre></div>
</div>
<p><a title="mpmath.acos" class="reference internal" href="#mpmath.acos"><tt class="xref docutils literal"><span class="pre">acos()</span></tt></a> is defined so as to be a proper inverse function of
<img class="math" src="../_images/math/42781e6e22b30f45c904fcccdaffa1124d6a2d77.png" alt="\cos(\theta)"/> for <img class="math" src="../_images/math/30087b20a659037731297991d312a356f34b51b8.png" alt="0 \le \theta < \pi"/>.
We have <img class="math" src="../_images/math/933d47d8f3212606cd793a36c87f02106a33a3c7.png" alt="\cos(\cos^{-1}(x)) = x"/> for all <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, but
<img class="math" src="../_images/math/60ccf6533f56deddecfab2df154dc80d132f4bdf.png" alt="\cos^{-1}(\cos(x)) = x"/> only for <img class="math" src="../_images/math/05d953a5eb14edccbb3765e404eca691e5f6b4d2.png" alt="0 \le \Re[x] < \pi"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="mi">10</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">]:</span>
<span class="gp">... </span> <span class="k">print</span><span class="p">(</span><span class="s">"</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">acos</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">acos</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
<span class="gp">...</span>
<span class="go">1.0 1.0</span>
<span class="go">(10.0 + 0.0j) 2.566370614359172953850574</span>
<span class="go">-1.0 1.0</span>
<span class="go">(2.0 + 3.0j) (2.0 + 3.0j)</span>
<span class="go">(10.0 + 3.0j) (2.566370614359172953850574 - 3.0j)</span>
</pre></div>
</div>
<p>The inverse cosine has two branch points: <img class="math" src="../_images/math/4f424f1753ba66a66c48a233c5afe322d41467a0.png" alt="x = \pm 1"/>. <a title="mpmath.acos" class="reference internal" href="#mpmath.acos"><tt class="xref docutils literal"><span class="pre">acos()</span></tt></a>
places the branch cuts along the line segments <img class="math" src="../_images/math/9f498d2ffdafc7d28bbba71b26b37477a7fe0868.png" alt="(-\infty, -1)"/> and
<img class="math" src="../_images/math/eb5a1e1dd7a49594090b2131d83e208c46d1f9b7.png" alt="(+1, +\infty)"/>. In general,</p>
<div class="math">
<p><img src="../_images/math/e701ac390a3291d5a972c8ccc6a1464aa790ca4f.png" alt="\cos^{-1}(x) = \frac{\pi}{2} + i \log\left(ix + \sqrt{1-x^2} \right)" /></p>
</div><p>where the principal-branch log and square root are implied.</p>
</dd></dl>
</div>
<div class="section" id="asin">
<h3><tt class="xref docutils literal"><span class="pre">asin()</span></tt><a class="headerlink" href="#asin" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.asin">
<tt class="descclassname">mpmath.</tt><tt class="descname">asin</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.asin" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse sine or arcsine of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/8edc8053b418a5136de3596ef2d02cc582677773.png" alt="\sin^{-1}(x)"/>.
Since <img class="math" src="../_images/math/4c5a79a38999632f998419f1ae27fd2d71b18208.png" alt="-1 \le \sin(x) \le 1"/> for real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, the inverse
sine is real-valued only for <img class="math" src="../_images/math/f260c78fb4c935f83297e9c2cb2cf7c3c3e3adfc.png" alt="-1 \le x \le 1"/>.
On this interval, it is defined to be a monotonically increasing
function assuming values between <img class="math" src="../_images/math/d45899b464b92ec7d9086f93e2ff440b344cfb55.png" alt="-\pi/2"/> and <img class="math" src="../_images/math/039bf945c704ff98e6f81626bbad32550cc37193.png" alt="\pi/2"/>.</p>
<p>Basic values are:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">asin</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">asin</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">asin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">asin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[0.0, 1.0, 0.0, 0.166667, 0.0, 0.075, 0.0]</span>
</pre></div>
</div>
<p><a title="mpmath.asin" class="reference internal" href="#mpmath.asin"><tt class="xref docutils literal"><span class="pre">asin()</span></tt></a> is defined so as to be a proper inverse function of
<img class="math" src="../_images/math/3ea19187c412da5e82c8ca0aacda5402d14ae7e2.png" alt="\sin(\theta)"/> for <img class="math" src="../_images/math/1bc41b3f5ce30d2cf0e5c41e3462065323d6792d.png" alt="-\pi/2 < \theta < \pi/2"/>.
We have <img class="math" src="../_images/math/0652bcfc2ad09ab1a7200a122ca1f5edee2690cb.png" alt="\sin(\sin^{-1}(x)) = x"/> for all <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, but
<img class="math" src="../_images/math/2b1ce7f6d5340aa14eae4db5b113e400c0b6fc23.png" alt="\sin^{-1}(\sin(x)) = x"/> only for <img class="math" src="../_images/math/74204fcd1d32ab2b5a7d5963f2fe82470746e23b.png" alt="-\pi/2 < \Re[x] < \pi/2"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">]:</span>
<span class="gp">... </span> <span class="k">print</span><span class="p">(</span><span class="s">"</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">))),</span> <span class="n">asin</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
<span class="gp">...</span>
<span class="go">1.0 1.0</span>
<span class="go">10.0 -0.5752220392306202846120698</span>
<span class="go">-1.0 -1.0</span>
<span class="go">(1.0 + 3.0j) (1.0 + 3.0j)</span>
<span class="go">(-2.0 + 3.0j) (-1.141592653589793238462643 - 3.0j)</span>
</pre></div>
</div>
<p>The inverse sine has two branch points: <img class="math" src="../_images/math/4f424f1753ba66a66c48a233c5afe322d41467a0.png" alt="x = \pm 1"/>. <a title="mpmath.asin" class="reference internal" href="#mpmath.asin"><tt class="xref docutils literal"><span class="pre">asin()</span></tt></a>
places the branch cuts along the line segments <img class="math" src="../_images/math/9f498d2ffdafc7d28bbba71b26b37477a7fe0868.png" alt="(-\infty, -1)"/> and
<img class="math" src="../_images/math/eb5a1e1dd7a49594090b2131d83e208c46d1f9b7.png" alt="(+1, +\infty)"/>. In general,</p>
<div class="math">
<p><img src="../_images/math/1a9d926d9add5ce55cfe0793749ce752bb2306dd.png" alt="\sin^{-1}(x) = -i \log\left(ix + \sqrt{1-x^2} \right)" /></p>
</div><p>where the principal-branch log and square root are implied.</p>
</dd></dl>
</div>
<div class="section" id="atan">
<h3><tt class="xref docutils literal"><span class="pre">atan()</span></tt><a class="headerlink" href="#atan" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.atan">
<tt class="descclassname">mpmath.</tt><tt class="descname">atan</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.atan" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse tangent or arctangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, <img class="math" src="../_images/math/b71ce9cd290d5a709487727440e4b29505c99309.png" alt="\tan^{-1}(x)"/>.
This is a real-valued function for all real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, with range
<img class="math" src="../_images/math/33fed50d641d8348a942f5d152134facaa22e6ad.png" alt="(-\pi/2, \pi/2)"/>.</p>
<p>Basic values are:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">-1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">-0.7853981633974483096156609</span>
<span class="gp">>>> </span><span class="n">atan</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.7853981633974483096156609</span>
<span class="gp">>>> </span><span class="n">atan</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">1.570796326794896619231322</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">atan</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
<span class="go">[0.0, 1.0, 0.0, -0.333333, 0.0, 0.2, 0.0]</span>
</pre></div>
</div>
<p>The inverse tangent is often used to compute angles. However,
the atan2 function is often better for this as it preserves sign
(see <a title="mpmath.atan2" class="reference internal" href="#mpmath.atan2"><tt class="xref docutils literal"><span class="pre">atan2()</span></tt></a>).</p>
<p><a title="mpmath.atan" class="reference internal" href="#mpmath.atan"><tt class="xref docutils literal"><span class="pre">atan()</span></tt></a> is defined so as to be a proper inverse function of
<img class="math" src="../_images/math/6c44768bb83598451c904bb196761ce4b1d1348f.png" alt="\tan(\theta)"/> for <img class="math" src="../_images/math/1bc41b3f5ce30d2cf0e5c41e3462065323d6792d.png" alt="-\pi/2 < \theta < \pi/2"/>.
We have <img class="math" src="../_images/math/563618bb10a5e9cc2ee74b186f6c3444c663d7f3.png" alt="\tan(\tan^{-1}(x)) = x"/> for all <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>, but
<img class="math" src="../_images/math/b5e209118b4e9bb13d5fb0fd66f775fbcbaf92aa.png" alt="\tan^{-1}(\tan(x)) = x"/> only for <img class="math" src="../_images/math/74204fcd1d32ab2b5a7d5963f2fe82470746e23b.png" alt="-\pi/2 < \Re[x] < \pi/2"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">]:</span>
<span class="gp">... </span> <span class="k">print</span><span class="p">(</span><span class="s">"</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">atan</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">atan</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
<span class="gp">...</span>
<span class="go">1.0 1.0</span>
<span class="go">10.0 0.5752220392306202846120698</span>
<span class="go">-1.0 -1.0</span>
<span class="go">(1.0 + 3.0j) (1.000000000000000000000001 + 3.0j)</span>
<span class="go">(-2.0 + 3.0j) (1.141592653589793238462644 + 3.0j)</span>
</pre></div>
</div>
<p>The inverse tangent has two branch points: <img class="math" src="../_images/math/63dcbd342180d4abe6948600ffc7305606b40201.png" alt="x = \pm i"/>. <a title="mpmath.atan" class="reference internal" href="#mpmath.atan"><tt class="xref docutils literal"><span class="pre">atan()</span></tt></a>
places the branch cuts along the line segments <img class="math" src="../_images/math/d8090a02c41ef2b54224a16d8cb3c14cfed57cae.png" alt="(-i \infty, -i)"/> and
<img class="math" src="../_images/math/b188655f5108667683e6de308ac83a0a3def08c9.png" alt="(+i, +i \infty)"/>. In general,</p>
<div class="math">
<p><img src="../_images/math/fe81327370a3e4a45ab6e1765a1cb80a1f67437d.png" alt="\tan^{-1}(x) = \frac{i}{2}\left(\log(1-ix)-\log(1+ix)\right)" /></p>
</div><p>where the principal-branch log is implied.</p>
</dd></dl>
</div>
<div class="section" id="atan2">
<h3><tt class="xref docutils literal"><span class="pre">atan2()</span></tt><a class="headerlink" href="#atan2" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.atan2">
<tt class="descclassname">mpmath.</tt><tt class="descname">atan2</tt><big>(</big><em>y</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.atan2" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the two-argument arctangent, <img class="math" src="../_images/math/9dc6015dd5641eb929664202273a3d3f879f9e51.png" alt="\mathrm{atan2}(y, x)"/>,
giving the signed angle between the positive <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>-axis and the
point <img class="math" src="../_images/math/adfd9ae8a3fac031d3b8b470a52a709a23d4d6d2.png" alt="(x, y)"/> in the 2D plane. This function is defined for
real <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> and <img class="math" src="../_images/math/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" alt="y"/> only.</p>
<p>The two-argument arctangent essentially computes
<img class="math" src="../_images/math/8dad02890d0db27747cedab53775c0d00798e9ff.png" alt="\mathrm{atan}(y/x)"/>, but accounts for the signs of both
<img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> and <img class="math" src="../_images/math/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" alt="y"/> to give the angle for the correct quadrant. The
following examples illustrate the difference:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">atan2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">1.</span><span class="p">)</span>
<span class="go">(0.785398163397448, 0.785398163397448)</span>
<span class="gp">>>> </span><span class="n">atan2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="o">/-</span><span class="mf">1.</span><span class="p">)</span>
<span class="go">(2.35619449019234, -0.785398163397448)</span>
<span class="gp">>>> </span><span class="n">atan2</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="mf">1.</span><span class="p">)</span>
<span class="go">(-0.785398163397448, -0.785398163397448)</span>
<span class="gp">>>> </span><span class="n">atan2</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/-</span><span class="mf">1.</span><span class="p">)</span>
<span class="go">(-2.35619449019234, 0.785398163397448)</span>
</pre></div>
</div>
<p>The angle convention is the same as that used for the complex
argument; see <a title="mpmath.arg" class="reference external" href="../general.html#mpmath.arg"><tt class="xref docutils literal"><span class="pre">arg()</span></tt></a>.</p>
</dd></dl>
</div>
<div class="section" id="asec">
<h3><tt class="xref docutils literal"><span class="pre">asec()</span></tt><a class="headerlink" href="#asec" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.asec">
<tt class="descclassname">mpmath.</tt><tt class="descname">asec</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.asec" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse secant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/8012cf7a70f89667ec4b2ebd2dac7e7b79f27f49.png" alt="\mathrm{sec}^{-1}(x) = \cos^{-1}(1/x)"/>.</dd></dl>
</div>
<div class="section" id="acsc">
<h3><tt class="xref docutils literal"><span class="pre">acsc()</span></tt><a class="headerlink" href="#acsc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acsc">
<tt class="descclassname">mpmath.</tt><tt class="descname">acsc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acsc" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse cosecant of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/3e8e2afb11a76e1bd602582a3529fdf36dfe7b82.png" alt="\mathrm{csc}^{-1}(x) = \sin^{-1}(1/x)"/>.</dd></dl>
</div>
<div class="section" id="acot">
<h3><tt class="xref docutils literal"><span class="pre">acot()</span></tt><a class="headerlink" href="#acot" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.acot">
<tt class="descclassname">mpmath.</tt><tt class="descname">acot</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acot" title="Permalink to this definition">¶</a></dt>
<dd>Computes the inverse cotangent of <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>,
<img class="math" src="../_images/math/bb8eb1280c23567e9f9c562a28fbbd692c0190d6.png" alt="\mathrm{cot}^{-1}(x) = \tan^{-1}(1/x)"/>.</dd></dl>
</div>
</div>
<div class="section" id="sinc-function">
<h2>Sinc function<a class="headerlink" href="#sinc-function" title="Permalink to this headline">¶</a></h2>
<div class="section" id="sinc">
<h3><tt class="xref docutils literal"><span class="pre">sinc()</span></tt><a class="headerlink" href="#sinc" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sinc">
<tt class="descclassname">mpmath.</tt><tt class="descname">sinc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sinc" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">sinc(x)</span></tt> computes the unnormalized sinc function, defined as</p>
<div class="math">
<p><img src="../_images/math/e7315df78c4286c99d922173f9185c3e050193a2.png" alt="\mathrm{sinc}(x) = \begin{cases}
\sin(x)/x, & \mbox{if } x \ne 0 \\
1, & \mbox{if } x = 0.
\end{cases}" /></p>
</div><p>See <a title="mpmath.sincpi" class="reference internal" href="#mpmath.sincpi"><tt class="xref docutils literal"><span class="pre">sincpi()</span></tt></a> for the normalized sinc function.</p>
<p>Simple values and limits include:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">sinc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">sinc</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.841470984807897</span>
<span class="gp">>>> </span><span class="n">sinc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>The integral of the sinc function is the sine integral Si:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">quad</span><span class="p">(</span><span class="n">sinc</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">0.946083070367183</span>
<span class="gp">>>> </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0.946083070367183</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sincpi">
<h3><tt class="xref docutils literal"><span class="pre">sincpi()</span></tt><a class="headerlink" href="#sincpi" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sincpi">
<tt class="descclassname">mpmath.</tt><tt class="descname">sincpi</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sincpi" title="Permalink to this definition">¶</a></dt>
<dd><p><tt class="docutils literal"><span class="pre">sincpi(x)</span></tt> computes the normalized sinc function, defined as</p>
<div class="math">
<p><img src="../_images/math/32eb18f6e7c15d7073009f205d58061ccaae9e06.png" alt="\mathrm{sinc}_{\pi}(x) = \begin{cases}
\sin(\pi x)/(\pi x), & \mbox{if } x \ne 0 \\
1, & \mbox{if } x = 0.
\end{cases}" /></p>
</div><p>Equivalently, we have
<img class="math" src="../_images/math/2ef0be79204eb80d936b974993f0e84ec0c5d8e8.png" alt="\mathrm{sinc}_{\pi}(x) = \mathrm{sinc}(\pi x)"/>.</p>
<p>The normalization entails that the function integrates
to unity over the entire real line:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">quadosc</span><span class="p">(</span><span class="n">sincpi</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mf">2.0</span><span class="p">)</span>
<span class="go">1.0</span>
</pre></div>
</div>
<p>Like, <a title="mpmath.sinpi" class="reference internal" href="#mpmath.sinpi"><tt class="xref docutils literal"><span class="pre">sinpi()</span></tt></a>, <a title="mpmath.sincpi" class="reference internal" href="#mpmath.sincpi"><tt class="xref docutils literal"><span class="pre">sincpi()</span></tt></a> is evaluated accurately
at its roots:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">sincpi</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
</dd></dl>
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<h3><a href="../index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference external" href="#">Trigonometric functions</a><ul>
<li><a class="reference external" href="#degree-radian-conversion">Degree-radian conversion</a><ul>
<li><a class="reference external" href="#degrees"><tt class="docutils literal"><span class="pre">degrees()</span></tt></a></li>
<li><a class="reference external" href="#radians"><tt class="docutils literal"><span class="pre">radians()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#id1">Trigonometric functions</a><ul>
<li><a class="reference external" href="#cos"><tt class="docutils literal"><span class="pre">cos()</span></tt></a></li>
<li><a class="reference external" href="#sin"><tt class="docutils literal"><span class="pre">sin()</span></tt></a></li>
<li><a class="reference external" href="#tan"><tt class="docutils literal"><span class="pre">tan()</span></tt></a></li>
<li><a class="reference external" href="#sec"><tt class="docutils literal"><span class="pre">sec()</span></tt></a></li>
<li><a class="reference external" href="#csc"><tt class="docutils literal"><span class="pre">csc()</span></tt></a></li>
<li><a class="reference external" href="#cot"><tt class="docutils literal"><span class="pre">cot()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#trigonometric-functions-with-modified-argument">Trigonometric functions with modified argument</a><ul>
<li><a class="reference external" href="#cospi"><tt class="docutils literal"><span class="pre">cospi()</span></tt></a></li>
<li><a class="reference external" href="#sinpi"><tt class="docutils literal"><span class="pre">sinpi()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#inverse-trigonometric-functions">Inverse trigonometric functions</a><ul>
<li><a class="reference external" href="#acos"><tt class="docutils literal"><span class="pre">acos()</span></tt></a></li>
<li><a class="reference external" href="#asin"><tt class="docutils literal"><span class="pre">asin()</span></tt></a></li>
<li><a class="reference external" href="#atan"><tt class="docutils literal"><span class="pre">atan()</span></tt></a></li>
<li><a class="reference external" href="#atan2"><tt class="docutils literal"><span class="pre">atan2()</span></tt></a></li>
<li><a class="reference external" href="#asec"><tt class="docutils literal"><span class="pre">asec()</span></tt></a></li>
<li><a class="reference external" href="#acsc"><tt class="docutils literal"><span class="pre">acsc()</span></tt></a></li>
<li><a class="reference external" href="#acot"><tt class="docutils literal"><span class="pre">acot()</span></tt></a></li>
</ul>
</li>
<li><a class="reference external" href="#sinc-function">Sinc function</a><ul>
<li><a class="reference external" href="#sinc"><tt class="docutils literal"><span class="pre">sinc()</span></tt></a></li>
<li><a class="reference external" href="#sincpi"><tt class="docutils literal"><span class="pre">sincpi()</span></tt></a></li>
</ul>
</li>
</ul>
</li>
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