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<a name="Trigonometry"></a>
<div class="header">
<p>
Next: <a href="Sums-and-Products.html#Sums-and-Products" accesskey="n" rel="next">Sums and Products</a>, Previous: <a href="Complex-Arithmetic.html#Complex-Arithmetic" accesskey="p" rel="prev">Complex Arithmetic</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Trigonometry-1"></a>
<h3 class="section">17.3 Trigonometry</h3>
<p>Octave provides the following trigonometric functions where angles are
specified in radians. To convert from degrees to radians multiply by
<code>pi/180</code>
or use the <code>deg2rad</code> function. For example, <code>sin (30 * pi/180)</code>
returns the sine of 30 degrees. As an alternative, Octave provides a number of
trigonometric functions which work directly on an argument specified in
degrees. These functions are named after the base trigonometric function with
a ‘<samp>d</samp>’ suffix. As an example, <code>sin</code> expects an angle in radians while
<code>sind</code> expects an angle in degrees.
</p>
<p>Octave uses the C library trigonometric functions. It is expected that these
functions are defined by the ISO/IEC 9899 Standard. This Standard is available
at: <a href="http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf">http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf</a>.
Section F.9.1 deals with the trigonometric functions. The behavior of most of
the functions is relatively straightforward. However, there are some
exceptions to the standard behavior. Many of the exceptions involve the
behavior for -0. The most complex case is atan2. Octave exactly implements
the behavior given in the Standard. Including
<code>atan2(+- 0, 0)</code> returns <code>+- pi</code>.
</p>
<p>It should be noted that <small>MATLAB</small> uses different definitions which apparently
do not distinguish -0.
</p>
<a name="XREFdeg2rad"></a><dl>
<dt><a name="index-deg2rad"></a><em><var>rad</var> =</em> <strong>deg2rad</strong> <em>(<var>deg</var>)</em></dt>
<dd>
<p>Convert degrees to radians.
</p>
<p>The input <var>deg</var> must be a scalar, vector, or N-dimensional array of
double or single floating point values. <var>deg</var> may be complex in which
case the real and imaginary components are converted separately.
</p>
<p>The output <var>rad</var> is the same size and shape as <var>deg</var> with degrees
converted to radians using the conversion constant <code>pi/180</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">deg2rad ([0, 90, 180, 270, 360])
⇒ 0.00000 1.57080 3.14159 4.71239 6.28319
</pre></div>
<p><strong>See also:</strong> <a href="#XREFrad2deg">rad2deg</a>.
</p></dd></dl>
<a name="XREFrad2deg"></a><dl>
<dt><a name="index-rad2deg"></a><em><var>deg</var> =</em> <strong>rad2deg</strong> <em>(<var>rad</var>)</em></dt>
<dd>
<p>Convert radians to degrees.
</p>
<p>The input <var>rad</var> must be a scalar, vector, or N-dimensional array of
double or single floating point values. <var>rad</var> may be complex in which
case the real and imaginary components are converted separately.
</p>
<p>The output <var>deg</var> is the same size and shape as <var>rad</var> with radians
converted to degrees using the conversion constant <code>180/pi</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">rad2deg ([0, pi/2, pi, 3/2*pi, 2*pi])
⇒ 0 90 180 270 360
</pre></div>
<p><strong>See also:</strong> <a href="#XREFdeg2rad">deg2rad</a>.
</p></dd></dl>
<a name="XREFsin"></a><dl>
<dt><a name="index-sin"></a><em></em> <strong>sin</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the sine for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFasin">asin</a>, <a href="#XREFsind">sind</a>, <a href="#XREFsinh">sinh</a>.
</p></dd></dl>
<a name="XREFcos"></a><dl>
<dt><a name="index-cos"></a><em></em> <strong>cos</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cosine for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFacos">acos</a>, <a href="#XREFcosd">cosd</a>, <a href="#XREFcosh">cosh</a>.
</p></dd></dl>
<a name="XREFtan"></a><dl>
<dt><a name="index-tan"></a><em></em> <strong>tan</strong> <em>(<var>z</var>)</em></dt>
<dd><p>Compute the tangent for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFatan">atan</a>, <a href="#XREFtand">tand</a>, <a href="#XREFtanh">tanh</a>.
</p></dd></dl>
<a name="XREFsec"></a><dl>
<dt><a name="index-sec"></a><em></em> <strong>sec</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the secant for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFasec">asec</a>, <a href="#XREFsecd">secd</a>, <a href="#XREFsech">sech</a>.
</p></dd></dl>
<a name="XREFcsc"></a><dl>
<dt><a name="index-csc"></a><em></em> <strong>csc</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cosecant for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFacsc">acsc</a>, <a href="#XREFcscd">cscd</a>, <a href="#XREFcsch">csch</a>.
</p></dd></dl>
<a name="XREFcot"></a><dl>
<dt><a name="index-cot"></a><em></em> <strong>cot</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cotangent for each element of <var>x</var> in radians.
</p>
<p><strong>See also:</strong> <a href="#XREFacot">acot</a>, <a href="#XREFcotd">cotd</a>, <a href="#XREFcoth">coth</a>.
</p></dd></dl>
<a name="XREFasin"></a><dl>
<dt><a name="index-asin"></a><em></em> <strong>asin</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse sine in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsin">sin</a>, <a href="#XREFasind">asind</a>.
</p></dd></dl>
<a name="XREFacos"></a><dl>
<dt><a name="index-acos"></a><em></em> <strong>acos</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cosine in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcos">cos</a>, <a href="#XREFacosd">acosd</a>.
</p></dd></dl>
<a name="XREFatan"></a><dl>
<dt><a name="index-atan"></a><em></em> <strong>atan</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse tangent in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFtan">tan</a>, <a href="#XREFatand">atand</a>.
</p></dd></dl>
<a name="XREFasec"></a><dl>
<dt><a name="index-asec"></a><em></em> <strong>asec</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse secant in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsec">sec</a>, <a href="#XREFasecd">asecd</a>.
</p></dd></dl>
<a name="XREFacsc"></a><dl>
<dt><a name="index-acsc"></a><em></em> <strong>acsc</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cosecant in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcsc">csc</a>, <a href="#XREFacscd">acscd</a>.
</p></dd></dl>
<a name="XREFacot"></a><dl>
<dt><a name="index-acot"></a><em></em> <strong>acot</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cotangent in radians for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcot">cot</a>, <a href="#XREFacotd">acotd</a>.
</p></dd></dl>
<a name="XREFsinh"></a><dl>
<dt><a name="index-sinh"></a><em></em> <strong>sinh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the hyperbolic sine for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFasinh">asinh</a>, <a href="#XREFcosh">cosh</a>, <a href="#XREFtanh">tanh</a>.
</p></dd></dl>
<a name="XREFcosh"></a><dl>
<dt><a name="index-cosh"></a><em></em> <strong>cosh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the hyperbolic cosine for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFacosh">acosh</a>, <a href="#XREFsinh">sinh</a>, <a href="#XREFtanh">tanh</a>.
</p></dd></dl>
<a name="XREFtanh"></a><dl>
<dt><a name="index-tanh"></a><em></em> <strong>tanh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute hyperbolic tangent for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFatanh">atanh</a>, <a href="#XREFsinh">sinh</a>, <a href="#XREFcosh">cosh</a>.
</p></dd></dl>
<a name="XREFsech"></a><dl>
<dt><a name="index-sech"></a><em></em> <strong>sech</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the hyperbolic secant of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFasech">asech</a>.
</p></dd></dl>
<a name="XREFcsch"></a><dl>
<dt><a name="index-csch"></a><em></em> <strong>csch</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the hyperbolic cosecant of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFacsch">acsch</a>.
</p></dd></dl>
<a name="XREFcoth"></a><dl>
<dt><a name="index-coth"></a><em></em> <strong>coth</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the hyperbolic cotangent of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFacoth">acoth</a>.
</p></dd></dl>
<a name="XREFasinh"></a><dl>
<dt><a name="index-asinh"></a><em></em> <strong>asinh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic sine for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsinh">sinh</a>.
</p></dd></dl>
<a name="XREFacosh"></a><dl>
<dt><a name="index-acosh"></a><em></em> <strong>acosh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic cosine for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcosh">cosh</a>.
</p></dd></dl>
<a name="XREFatanh"></a><dl>
<dt><a name="index-atanh"></a><em></em> <strong>atanh</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic tangent for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFtanh">tanh</a>.
</p></dd></dl>
<a name="XREFasech"></a><dl>
<dt><a name="index-asech"></a><em></em> <strong>asech</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic secant of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsech">sech</a>.
</p></dd></dl>
<a name="XREFacsch"></a><dl>
<dt><a name="index-acsch"></a><em></em> <strong>acsch</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic cosecant of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcsch">csch</a>.
</p></dd></dl>
<a name="XREFacoth"></a><dl>
<dt><a name="index-acoth"></a><em></em> <strong>acoth</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse hyperbolic cotangent of each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcoth">coth</a>.
</p></dd></dl>
<a name="XREFatan2"></a><dl>
<dt><a name="index-atan2"></a><em></em> <strong>atan2</strong> <em>(<var>y</var>, <var>x</var>)</em></dt>
<dd><p>Compute atan (<var>y</var> / <var>x</var>) for corresponding elements of <var>y</var>
and <var>x</var>.
</p>
<p><var>y</var> and <var>x</var> must match in size and orientation. The signs of
elements of <var>y</var> and <var>x</var> are used to determine the quadrants of each
resulting value.
</p>
<p>This function is equivalent to <code>arg (complex (<var>x</var>, <var>y</var>))</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFtan">tan</a>, <a href="#XREFtand">tand</a>, <a href="#XREFtanh">tanh</a>, <a href="#XREFatanh">atanh</a>.
</p></dd></dl>
<p>Octave provides the following trigonometric functions where angles are
specified in degrees. These functions produce true zeros at the appropriate
intervals rather than the small round-off error that occurs when using
radians. For example:
</p>
<div class="example">
<pre class="example">cosd (90)
⇒ 0
cos (pi/2)
⇒ 6.1230e-17
</pre></div>
<a name="XREFsind"></a><dl>
<dt><a name="index-sind"></a><em></em> <strong>sind</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the sine for each element of <var>x</var> in degrees.
</p>
<p>Returns zero for elements where <code><var>x</var>/180</code> is an integer.
</p>
<p><strong>See also:</strong> <a href="#XREFasind">asind</a>, <a href="#XREFsin">sin</a>.
</p></dd></dl>
<a name="XREFcosd"></a><dl>
<dt><a name="index-cosd"></a><em></em> <strong>cosd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cosine for each element of <var>x</var> in degrees.
</p>
<p>Returns zero for elements where <code>(<var>x</var>-90)/180</code> is an integer.
</p>
<p><strong>See also:</strong> <a href="#XREFacosd">acosd</a>, <a href="#XREFcos">cos</a>.
</p></dd></dl>
<a name="XREFtand"></a><dl>
<dt><a name="index-tand"></a><em></em> <strong>tand</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the tangent for each element of <var>x</var> in degrees.
</p>
<p>Returns zero for elements where <code><var>x</var>/180</code> is an integer and
<code>Inf</code> for elements where <code>(<var>x</var>-90)/180</code> is an integer.
</p>
<p><strong>See also:</strong> <a href="#XREFatand">atand</a>, <a href="#XREFtan">tan</a>.
</p></dd></dl>
<a name="XREFsecd"></a><dl>
<dt><a name="index-secd"></a><em></em> <strong>secd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the secant for each element of <var>x</var> in degrees.
</p>
<p><strong>See also:</strong> <a href="#XREFasecd">asecd</a>, <a href="#XREFsec">sec</a>.
</p></dd></dl>
<a name="XREFcscd"></a><dl>
<dt><a name="index-cscd"></a><em></em> <strong>cscd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cosecant for each element of <var>x</var> in degrees.
</p>
<p><strong>See also:</strong> <a href="#XREFacscd">acscd</a>, <a href="#XREFcsc">csc</a>.
</p></dd></dl>
<a name="XREFcotd"></a><dl>
<dt><a name="index-cotd"></a><em></em> <strong>cotd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the cotangent for each element of <var>x</var> in degrees.
</p>
<p><strong>See also:</strong> <a href="#XREFacotd">acotd</a>, <a href="#XREFcot">cot</a>.
</p></dd></dl>
<a name="XREFasind"></a><dl>
<dt><a name="index-asind"></a><em></em> <strong>asind</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse sine in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsind">sind</a>, <a href="#XREFasin">asin</a>.
</p></dd></dl>
<a name="XREFacosd"></a><dl>
<dt><a name="index-acosd"></a><em></em> <strong>acosd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cosine in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcosd">cosd</a>, <a href="#XREFacos">acos</a>.
</p></dd></dl>
<a name="XREFatand"></a><dl>
<dt><a name="index-atand"></a><em></em> <strong>atand</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse tangent in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFtand">tand</a>, <a href="#XREFatan">atan</a>.
</p></dd></dl>
<a name="XREFatan2d"></a><dl>
<dt><a name="index-atan2d"></a><em></em> <strong>atan2d</strong> <em>(<var>y</var>, <var>x</var>)</em></dt>
<dd><p>Compute atan (<var>y</var> / <var>x</var>) in degrees for corresponding elements
from <var>y</var> and <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFtand">tand</a>, <a href="#XREFatan2">atan2</a>.
</p></dd></dl>
<a name="XREFasecd"></a><dl>
<dt><a name="index-asecd"></a><em></em> <strong>asecd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse secant in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFsecd">secd</a>, <a href="#XREFasec">asec</a>.
</p></dd></dl>
<a name="XREFacscd"></a><dl>
<dt><a name="index-acscd"></a><em></em> <strong>acscd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cosecant in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcscd">cscd</a>, <a href="#XREFacsc">acsc</a>.
</p></dd></dl>
<a name="XREFacotd"></a><dl>
<dt><a name="index-acotd"></a><em></em> <strong>acotd</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the inverse cotangent in degrees for each element of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcotd">cotd</a>, <a href="#XREFacot">acot</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Sums-and-Products.html#Sums-and-Products" accesskey="n" rel="next">Sums and Products</a>, Previous: <a href="Complex-Arithmetic.html#Complex-Arithmetic" accesskey="p" rel="prev">Complex Arithmetic</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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