Trigonometry (from Greek trigonon "triangle" and metron "measure") is a
branch of mathematics that studies triangles and the relationships between their sides
and the angles between the sides. Trigonometry also defines the trigonometric functions,
which describe those relationships.
Plane trigonometry deals with plane triangles of which the three vertices
are located on the surface of a plane. Spherical trigonometry deals with
spherical triangles of which the sides and vertices are located on the surface
of a sphere.
In navigation, common problems such as distance between two locations or
celestial sight reduction can be resolved by solving the values of sides and
angles of spherical triangles.
For these navigational problems, the Earth can be assumed to be a sphere
without introducing significant errors.
The next sections give an overview of the basic trigonometric relations
and identities, which may be usefull for theoretical work on plane and
spherical trigonometry. Further, some applications dealing with practical
navigational problems are discussed.
In the following sections, angles are assumed to be expressed
in degrees. Also the basic trigonometric functions Sine (sin(x))
and Cosine (cos(x)) are assumed to take their argument (x) in degrees.
This also inplies that the reverse functions Arcsine (asin(x)) and Arccosine
(acos(x)) return angle values in degrees.
Some calculators will accept angle values only in radians. An angle
value "xdegrees" expressed in degrees is converted
into an angle value "xradians" expressed in radians
by the following identity:
xradians = xdegrees
* 3.1415926/180 [radians/degrees].
The reverse conversion is given by:
xdegrees = xradians
* 180/3.1415926 [degrees/radians]
In some of the folowing sections, the function atan2() is used.
This is a modified version of the Arctangens function, which takes two arguments instead
of one argument.
The two arguments of the atan2() function are processed as follows:
atan2(y,x) = atan(y/x).
Viewed in terms of a Cartesian coordinate system, the result of the atan2(y,x) function
is the angle between the positive x-axis and a line drawn from the origin through
the point (x,y), using the signs of both arguments x and y to determine the quadrant of the
returned angle. this way,the returned angle can be correctly calculated over a full circle:
from -180° to +180°, whereas the simple atan() function may have an ambiguity of
180° on the returned value.
atan2(y,x) = atan(y/x) for x > 0
atan2(y,x) = atan(y/x) + 180° for x < 0 and y > 0
atan2(y,x) = atan(y/x) - 180° for x < 0 and y < 0