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📖 NAME

Math::Trig - trigonometric functions

🚀 Quick Reference

Use Case	Command	Description
🔄 Basic trig (tan, acos, asin)	tan(0.9), acos(3.7), asin(2.4)	Extended trig functions beyond sin/cos
🔢 Constant pi	pi/2, pi/4	Pi and multiples (pip2, pip4, pi2)
🌡️ Angle conversion	deg2rad(120)	Degrees ↔ radians ↔ gradians
🌐 Radial coordinate conversion	cartesian_to_spherical($x,$y,$z)	Cartesian ↔ spherical ↔ cylindrical
🌍 Great circle distance	great_circle_distance($theta0,$phi0,$theta1,$phi1)	Shortest distance between two points on a sphere
🧭 Great circle bearing	great_circle_direction($theta0,$phi0,$theta1,$phi1)	Initial direction (bearing) for great circle route
📍 Great circle destination	great_circle_destination($theta,$phi,$diro,$distance)	Compute destination from start, bearing, distance
📐 Real-valued asin/acos	asin_real($val), acos_real($val)	Guaranteed real output for inputs in [-1,1]

📝 SYNOPSIS

use Math::Trig;

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

$halfpi = pi/2;

$rad = deg2rad(120);

# Import constants pi2, pip2, pip4 (2*pi, pi/2, pi/4).
use Math::Trig ':pi';

# Import the conversions between cartesian/spherical/cylindrical.
use Math::Trig ':radial';

# Import the great circle formulas.
use Math::Trig ':great_circle';

📖 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, and great circle formulas for spherical movement.

📐 ANGLES

All angles are defined in radians, except where otherwise specified (for example in the deg/rad conversion functions).

📐 TRIGONOMETRIC FUNCTIONS

• Tangent: tan
• Cofunctions: csc, cosec, sec, sec, cot, cotan (cosec/csc and cotan/cot are aliases)
• Arcus (inverse) functions: asin, acos, atan
• Principal value of arc tangent of y/x: atan2(y, x)
• Arcus cofunctions: acsc, acosec, asec, acot, acotan (acosec/acsc and acotan/acot are aliases). Note that atan2(0, 0) is not well-defined.
• Hyperbolic functions: sinh, cosh, tanh
• Hyperbolic cofunctions: csch, cosech, sech, coth, cotanh (cosech/csch and cotanh/coth are aliases)
• Area (inverse hyperbolic) functions: asinh, acosh, atanh
• Area cofunctions: acsch, acosech, asech, acoth, acotanh
• Constants: pi, pi2, pi4, pip2, pip4

⚠️ ERRORS DUE TO DIVISION BY ZERO

The following functions cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of zero. These situations cause fatal runtime errors:

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

Example errors:

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

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

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

Note that atan2(0, 0) is not well-defined.

🔢 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 perldata) 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 Math::Complex for more information. In practice you need not to worry about 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.

• deg2rad: $radians = deg2rad($degrees);
• grad2rad: $radians = grad2rad($gradians);
• rad2deg: $degrees = rad2deg($radians);
• grad2deg: $degrees = grad2deg($gradians);
• deg2grad: $gradians = deg2grad($degrees);
• rad2grad: $gradians = rad2grad($radians);

The full circle is 2 pi radians or 360 degrees or 400 gradians. The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. If you don't want this, supply a true second argument:

$zillions_of_radians  = deg2rad($zillions_of_degrees, 1);
$negative_degrees     = rad2deg($negative_radians, 1);

You can also do the wrapping explicitly by rad2rad(), deg2deg(), and grad2grad().

• rad2rad: $radians_wrapped_by_2pi = rad2rad($radians);
• deg2deg: $degrees_wrapped_by_360 = deg2deg($degrees);
• grad2grad: $gradians_wrapped_by_400 = grad2grad($gradians);

🌐 RADIAL COORDINATE CONVERSIONS

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 :radial tag:

use Math::Trig ':radial';

($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);

All angles are in radians.

COORDINATE SYSTEMS

• Cartesian coordinates: usual rectangular (x, y, z)-coordinates.
• Spherical coordinates: (rho, theta, pi) – three-dimensional coordinates based on a sphere surface. The radius is rho (radial), the angle in the xy-plane is theta (azimuthal), the angle from the z-axis is phi (polar). The North Pole is 0, 0, rho; the Gulf of Guinea is 0, pi/2, rho. In geographical terms phi is latitude (northward positive, southward negative) and theta is longitude (eastward positive, westward negative). BEWARE: some texts define theta and phi the other way round, some define phi from the horizontal plane, some use r in place of rho.
• Cylindrical coordinates: (rho, theta, z) – based on a cylinder surface. The radius is rho (radial), the angle in the xy-plane is theta (azimuthal), the third coordinate is 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 AND DIRECTIONS

A great circle is section of a circle that contains the circle diameter: the shortest distance between two (non-antipodal) points on the spherical surface goes along the great circle connecting those two points.

great_circle_distance

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, $phi1, [, $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 radians.

If you think geographically the theta are longitudes: zero at the Greenwhich meridian, eastward positive, westward negative -- and the phi are latitudes: zero at the North Pole, northward positive, southward negative. NOTE: this formula thinks in mathematics, not geographically: the phi zero is at the North Pole, not at the Equator on the west coast of Africa (Bay of Guinea). You need to subtract your geographical coordinates from pi/2 (also known as 90 degrees).

$distance = great_circle_distance($lon0, pi/2 - $lat0,
                                  $lon1, pi/2 - $lat1, $rho);

great_circle_direction

The direction you must follow the great circle (also known as bearing) can be computed by the great_circle_direction() function:

use Math::Trig 'great_circle_direction';

$direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);

great_circle_bearing

Alias 'great_circle_bearing' for 'great_circle_direction' is also available.

use Math::Trig 'great_circle_bearing';

$direction = great_circle_bearing($theta0, $phi0, $theta1, $phi1);

The result of great_circle_direction is in radians, zero indicating straight north, pi or -pi straight south, pi/2 straight west, and -pi/2 straight east.

great_circle_destination

You can inversely compute the destination if you know the starting point, direction, and distance:

use Math::Trig 'great_circle_destination';

# $diro is the original direction,
# for example from great_circle_bearing().
# $distance is the angular distance in radians,
# for example from great_circle_distance().
# $thetad and $phid are the destination coordinates,
# $dird is the final direction at the destination.

($thetad, $phid, $dird) =
  great_circle_destination($theta, $phi, $diro, $distance);

or the midpoint if you know the end points:

great_circle_midpoint

use Math::Trig 'great_circle_midpoint';

($thetam, $phim) =
  great_circle_midpoint($theta0, $phi0, $theta1, $phi1);

The great_circle_midpoint() is just a special case (with $way = 0.5) of

great_circle_waypoint

use Math::Trig 'great_circle_waypoint';

($thetai, $phii) =
  great_circle_waypoint($theta0, $phi0, $theta1, $phi1, $way);

Where the $way is a value from zero ($theta0, $phi0) to one ($theta1, $phi1). Note that antipodal points (where their distance is pi radians) do not have waypoints between them (they would have an an "equator" between them), and therefore "undef" is returned for antipodal points. If the points are the same and the distance therefore zero and all waypoints therefore identical, the first point (either point) is returned.

The thetas, phis, direction, and distance in the above are all in radians.

You can import all the great circle formulas by

use Math::Trig ':great_circle';

Notice that the resulting directions might be somewhat surprising if you are looking at a flat worldmap: in such map projections the great circles quite often do not look like the shortest routes -- but for example the shortest possible routes from Europe or North America to Asia do often cross the polar regions. (The common Mercator projection does not show great circles as straight lines: straight lines in the Mercator projection are lines of constant bearing.)

📝 EXAMPLES

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

use Math::Trig qw(great_circle_distance deg2rad);

# Notice the 90 - latitude: phi zero is at the North Pole.
sub NESW { deg2rad($_[0]), deg2rad(90 - $_[1]) }
my @L = NESW( -0.5, 51.3);
my @T = NESW(139.8, 35.7);
my $km = great_circle_distance(@L, @T, 6378); # About 9600 km.

The direction you would have to go from London to Tokyo (in radians, straight north being zero, straight east being pi/2).

use Math::Trig qw(great_circle_direction);

my $rad = great_circle_direction(@L, @T); # About 0.547 or 0.174 pi.

The midpoint between London and Tokyo being

use Math::Trig qw(great_circle_midpoint);

my @M = great_circle_midpoint(@L, @T);

or about 69 N 89 E, in the frozen wastes of Siberia.

NOTE: you cannot get from A to B like this:

Dist = great_circle_distance(A, B)
Dir  = great_circle_direction(A, B)
C    = great_circle_destination(A, Dist, Dir)

and expect C to be B, because the bearing constantly changes when going from A to B (except in some special case like the meridians or the circles of latitudes) and in great_circle_destination() one gives a constant bearing to follow.

CAVEAT FOR GREAT CIRCLE FORMULAS

The answers may be off by few percentages because of the irregular (slightly aspherical) form of the Earth. The errors are at worst about 0.55%, but generally below 0.3%.

Real-valued asin and acos

For small inputs asin() and acos() may return complex numbers even when real numbers would be enough and correct, this happens because of floating-point inaccuracies. You can see these inaccuracies for example by trying theses:

print cos(1e-6)**2+sin(1e-6)**2 - 1,"\n";
printf "%.20f", cos(1e-6)**2+sin(1e-6)**2,"\n";

which will print something like this

-1.11022302462516e-16
0.99999999999999988898

even though the expected results are of course exactly zero and one. The formulas used to compute asin() and acos() are quite sensitive to this, and therefore they might accidentally slip into the complex plane even when they should not. To counter this there are two interfaces that are guaranteed to return a real-valued output.

• asin_real:

  use Math::Trig qw(asin_real);
  
  $real_angle = asin_real($input_sin);

  Return a real-valued arcus sine if the input is between [-1, 1], inclusive the endpoints. For inputs greater than one, pi/2 is returned. For inputs less than minus one, -pi/2 is returned.
• acos_real:

  use Math::Trig qw(acos_real);
  
  $real_angle = acos_real($input_cos);

  Return a real-valued arcus cosine if the input is between [-1, 1], inclusive the endpoints. For inputs greater than one, zero is returned. For inputs less than minus one, pi is returned.

🐛 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.

Do not attempt navigation using these formulas.

📚 SEE ALSO

Math::Complex

👤 AUTHORS

Jarkko Hietaniemi <jhi!at!iki.fi>, Raphael Manfredi <Raphael_Manfredi!at!pobox.com>, Zefram <zefram AT fysh.org>

📄 LICENSE

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

perl v5.34.0 2026-06-23 Math::Trig(3perl)

📖 NAME 🚀 Quick Reference 📝 SYNOPSIS 📖 DESCRIPTION 📐 ANGLES 📐 TRIGONOMETRIC FUNCTIONS 🔄 PLANE ANGLE CONVERSIONS 🌐 RADIAL COORDINATE CONVERSIONS 🌍 GREAT CIRCLE DISTANCES AND DIRECTIONS 📝 EXAMPLES 🐛 BUGS 📚 SEE ALSO 👤 AUTHORS 📄 LICENSE