Spherical trigonometry , unlike the plane trigonometry, deals with relation between the sides and angles of spherical triangles.
The spherical triangle is a triangle discribed by the arcs of great circles on the sphere. The great circles on a sphere are the circles whose centre is same as the centre of the sphere.
Let ABC be the spherical triangle with vertices A,B and C, formed by the great cirle's arc AB, arc BC and arc CA. Let arc AB=c, arc BC=a and arc AC = b.
Then the relation between the sides, a, b and c and angles A, B and C are given by sin rule and cosine rules:
Cos a = cos b*cos c +sin b*sin c cos A, which gives the side a interms of the other two sides and the included angle.
Sine rule: As in plane trigonometry, in spherical trigonometry also we a similar relationship. The sine rule could be derived from the cosine rule also.
sine a / sine A = sin b / sine B = sin c/ sine C.
Spherical trigonometry is used in astronomy. The angular apparent positions of the planets and stars with respect each other, with respect to horizon and ecliptic could be dealt with using spherical trigonometry.
Spherical trigonometry mathematics is the discipline which deals with solving triangles formed on the surface of a sphere from arcs of large circles.
Spherical trigonometry has a big importance, theoretical and practical, and it's applicable on a larger scale in astronomy, superior geodesy, in cartography, in crystallography, in the mining geometry, theory of instruments and other sciences, when, to study the relative position in space of points, lines and planes, it is used an helpful sphere.
It's called spherical surface, or area, the geometrical locus of points in space, equal far off from a fixed point O - the center of this surface.
The space bounded by the surface of a sphere is called also sphere.
Spherical surface can be defined as the surface produced by rotating a semicircle around it's diameter.
Segment of straight line joining the center of the sphere with any point on its surface is called the sphere radius R and the segment of straight line , joining two points on the spherical surface passing through its center and is called diameter, obviously the same sphere radii are equal between them and the diameter is equal with two rays.
In the spherical geometry are used the following theorems:
Theorem 1: Section of a sphere with a certain plane, is a circle.
Theorem 2: great circles divide the sphere and its surface into two equal parts.
Theorem 3: Through two given points on the surface of a sphere, if they are not placed at the ends of the same diameter, it can be traced a large circle and only one.
Theorem 4: The intersection plane of two large circles is one of their diameter and divides them into two equal parts.
Theorem 5: The shortest distance between two points on the sphere, on its surface, is an large arc smaller than 180 °.