Trigonometry

Trigonometry' (from Greek trigōnon ``triangle + metron ``measure) is a branch of mathematics that studies the relationship of lengths and angles in triangles. The field emerged during the third century BC, evolving out of a understanding of geometry then being used extensively for astronomical studies .

The 3rd century astronomers first noted that the lengths of the sides of a right angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier Transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across a great many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology and biology. Trigonometry is also the foundation of the practical art of surveying.

Trigonometry is most simply associated with planar right angle triangles (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 relates to right angle triangles. One exceptions 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.

Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course.

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: