# Math::Trig

```Math::Trig(3)          Perl Programmers Reference Guide          Math::Trig(3)

NAME
Math::Trig - trigonometric functions

SYNOPSIS
use Math::Trig;

\$x = tan(0.9);
\$y = acos(3.7);
\$z = asin(2.4);

\$halfpi = pi/2;

DESCRIPTION
Math::Trig defines many trigonometric functions not defined by the core
Perl which defines only the sin() and cos().  The constant pi is also
defined as are a few convenience functions for angle conversions.

TRIGONOMETRIC FUNCTIONS
The tangent

tan

The cofunctions of the sine, cosine, and tangent (cosec/csc and
cotan/cot are aliases)

csc, cosec, sec, sec, cot, cotan

The arcus (also known as the inverse) functions of the sine, cosine,
and tangent

asin, acos, atan

The principal value of the arc tangent of y/x

atan2(y, x)

The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc and
acotan/acot are aliases)

acsc, acosec, asec, acot, acotan

The hyperbolic sine, cosine, and tangent

sinh, cosh, tanh

The cofunctions of the hyperbolic sine, cosine, and tangent
(cosech/csch and cotanh/coth are aliases)

csch, cosech, sech, coth, cotanh

The arcus (also known as the inverse) functions of the hyperbolic sine,
cosine, and tangent

asinh, acosh, atanh

The arcus cofunctions of the hyperbolic sine, cosine, and tangent
(acsch/acosech and acoth/acotanh are aliases)

acsch, acosech, asech, acoth, acotanh

The trigonometric constant pi is also defined.

\$pi2 = 2 * pi;

ERRORS DUE TO DIVISION BY ZERO

The following functions

acoth
acsc
acsch
asec
asech
atanh
cot
coth
csc
csch
sec
sech
tan
tanh

cannot be computed for all arguments because that would mean dividing
by zero or taking logarithm of zero. These situations cause fatal
runtime errors looking like this

cot(0): Division by zero.
(Because in the definition of cot(0), the divisor sin(0) is 0)
Died at ...

or

atanh(-1): Logarithm of zero.
Died at...

For the csc, cot, asec, acsc, acot, csch, coth, asech, acsch, the
argument cannot be 0 (zero).  For the atanh, acoth, the argument cannot
be 1 (one).  For the atanh, acoth, the argument cannot be -1 (minus
one).  For the tan, sec, tanh, sech, the argument cannot be pi/2 + k *
pi, where k is any integer.

SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS

Please note that some of the trigonometric functions can break out from
the real axis into the complex plane. For example asin(2) has no
definition for plain real numbers but it has definition for complex
numbers.

In Perl terms this means that supplying the usual Perl numbers (also
known as scalars, please see the perldata manpage) as input for the
trigonometric functions might produce as output results that no more
are simple real numbers: instead they are complex numbers.

The Math::Trig handles this by using the Math::Complex package which
knows how to handle complex numbers, please see the Math::Complex
getting complex numbers as results because the Math::Complex takes care
of details like for example how to display complex numbers. For
example:

print asin(2), "\n";

should produce something like this (take or leave few last decimals):

1.5707963267949-1.31695789692482i

That is, a complex number with the real part of approximately 1.571 and
the imaginary part of approximately -1.317.

PLANE ANGLE CONVERSIONS
(Plane, 2-dimensional) angles may be converted with the following
functions.

The full circle is 2 pi radians or 360 degrees or 400 gradians.

Radial coordinate systems are the spherical and the cylindrical
systems, explained shortly in more detail.

You can import radial coordinate conversion functions by using the

(\$rho, \$theta, \$z)     = cartesian_to_cylindrical(\$x, \$y, \$z);
(\$rho, \$theta, \$phi)   = cartesian_to_spherical(\$x, \$y, \$z);
(\$x, \$y, \$z)           = cylindrical_to_cartesian(\$rho, \$theta, \$z);
(\$rho_s, \$theta, \$phi) = cylindrical_to_spherical(\$rho_c, \$theta, \$z);
(\$x, \$y, \$z)           = spherical_to_cartesian(\$rho, \$theta, \$phi);
(\$rho_c, \$theta, \$z)   = spherical_to_cylindrical(\$rho_s, \$theta, \$phi);

COORDINATE SYSTEMS

Cartesian coordinates are the usual rectangular (x, y, z)-coordinates.

Spherical coordinates, (rho, theta, pi), are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a sphere surface.  The radius of the sphere is rho, also known
as the radial coordinate.  The angle in the xy-plane (around the
z-axis) is theta, also known as the azimuthal coordinate.  The angle
from the z-axis is phi, also known as the polar coordinate.  The `North
Pole' is therefore 0, 0, rho, and the `Bay of Guinea' (think of the
missing big chunk of Africa) 0, pi/2, rho.

Beware: some texts define theta and phi the other way round, some texts
define the phi to start from the horizontal plane, some texts use r in
place of rho.

Cylindrical coordinates, (rho, theta, z), are three-dimensional
coordinates which define a point in three-dimensional space.  They are
based on a cylinder surface.  The radius of the cylinder is rho, also
known as the radial coordinate.  The angle in the xy-plane (around the
z-axis) is theta, also known as the azimuthal coordinate.  The third
coordinate is the z, pointing up from the theta-plane.

3-D ANGLE CONVERSIONS

Conversions to and from spherical and cylindrical coordinates are
available.  Please notice that the conversions are not necessarily
reversible because of the equalities like pi angles being equal to -pi
angles.

cartesian_to_cylindrical

(\$rho, \$theta, \$z) = cartesian_to_cylindrical(\$x, \$y, \$z);

cartesian_to_spherical

(\$rho, \$theta, \$phi) = cartesian_to_spherical(\$x, \$y, \$z);

cylindrical_to_cartesian

(\$x, \$y, \$z) = cylindrical_to_cartesian(\$rho, \$theta, \$z);

cylindrical_to_spherical

(\$rho_s, \$theta, \$phi) = cylindrical_to_spherical(\$rho_c, \$theta, \$z);

Notice that when \$z is not 0 \$rho_s is not equal to \$rho_c.

spherical_to_cartesian

(\$x, \$y, \$z) = spherical_to_cartesian(\$rho, \$theta, \$phi);

spherical_to_cylindrical

(\$rho_c, \$theta, \$z) = spherical_to_cylindrical(\$rho_s, \$theta, \$phi);

Notice that when \$z is not 0 \$rho_c is not equal to \$rho_s.

GREAT CIRCLE DISTANCES
You can compute spherical distances, called great circle distances, by
importing the great_circle_distance function:

use Math::Trig 'great_circle_distance'

\$distance = great_circle_distance(\$theta0, \$phi0, \$theta1, \$phi, [, \$rho]);

The great circle distance is the shortest distance between two points
on a sphere.  The distance is in \$rho units.  The \$rho is optional, it
defaults to 1 (the unit sphere), therefore the distance defaults to

EXAMPLES
To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N
139.8E) in kilometers:

# Notice the 90 - latitude: phi zero is at the North Pole.

\$km = great_circle_distance(@L, @T, 6378);

The answer may be off by up to 0.3% because of the irregular (slightly
aspherical) form of the Earth.

BUGS
Saying use Math::Trig; exports many mathematical routines in the caller
environment and even overrides some (sin, cos).  This is construed as a
feature by the Authors, actually... ;-)

The code is not optimized for speed, especially because we use
Math::Complex and thus go quite near complex numbers while doing the
computations even when the arguments are not. This, however, cannot be
completely avoided if we want things like asin(2) to give an answer
instead of giving a fatal runtime error.

AUTHORS
Jarkko Hietaniemi <jhi@iki.fi> and Raphael Manfredi
<Raphael_Manfredi@grenoble.hp.com>.

3rd Berkeley Distribution    perl 5.005, patch 02                Math::Trig(3)```