Trigonometry

Trigonometry (from Greek trigōnon, “triangle” and metron, “measure”) is a branch of mathematics that studies relationships involving lengths and angles of triangles.
The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. 
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees.
The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles.
Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle.
If the length of one of the sides is known, the other two are determined.
These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

 

 

Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees).
The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles,
most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.
One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature,
in elliptic geometry (a fundamental part of astronomy and navigation). Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.