### TRIGONOMETRY

Trigonometry is the branch of mathematics that deals with the relationships
between the sides and angles of triangles.
PLANE TRIGONOMETRY

Plane trigonometry is the trigonometry of triangles in a plane (ie 2D).

*Scalene triangle in a plane*
This triangle is defined by three connected line segments. The vertices
are labelled A, B and C, and these symbols are also used to label the interior
angles at these vertices. The sides opposite each angle have a linear
dimension labelled by the corresponding lowercase letter (ie side a is
opposite angle A). The three interior angles sum to 180 degrees:

There are two fundamental laws relating the sides and angles:
When one of the angles (eg C) is equal to 90 degrees, we have a right-angled
triangle.

*Planar right triangle*
The above two laws then essentially reduce to basic definitions of
the fundamental trigonometrical functions:

SPHERICAL TRIGONOMETRY

Spherical trigonometry is the trigonometry of triangles on the surface
of a sphere (ie 3-D)

*General triangle on the surface of a sphere*
A spherical triangle is a triangle existing on the surface of a sphere,
defined by line segments joining three points A, B and C. The line segments
are the shortest distances on the sphere between their respective end points,
and as a result are portions of great circles with their centres at the
centre of the sphere.

As with plane trigonometry, the symbols A, B and C are also used to label
the angles between the line segments (or the planes containing the line
segments) on the surface of the sphere. However, unlike plane trigonometry
the sides a, b and c are not linear distances, but angles. These angles
are the angles subtended by the respective line segments at the center of
the sphere. In a spherical triangle the angles A, B and C do not sum to 180
degrees.

There are also two fundamental laws of spherical trigonometry:

When one of the included angles (eg C) is equal to 90 degrees, we have a
spherical right triangle.

*Spherical triangle with one right angle*
The law of sines then reduces to:

sin a = sin A sin c

sin b = sin B sin c

and the law of cosines reduces to:
APPLICATIONS

Both plane and spherical trigonometry play an essential role in the
solution of many problems involving astronomy, space navigation, geodesy,
surveillance, spacecraft design and planetary science.