Trigonometry, from the Greek trigonon, meaning triangle, and metron, meaning measure, is a branch of mathematics that studies the relationships involving lengths and angles of triangles. The field came about during the 3rd century BC from applications of geometry to astronomical studies.
The 3rd century astronomers first noted that the lengths of the sides of a right triangle and the angles between those sides have fixed relationships: that is, if at least the length of one of those sides 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 currently they are pervasive in both applied and pure mathematics: fundamental methods of analysis use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, astronomy, music and acoustics, ecology, and biology. Trigonometry is also the groundwork of the practical art of surveying.
Trigonometry is simply associated with planar right angle triangles. The applicability to non-right triangles exists but, since any non-right angle triangle can be bisected to create two right angle triangles, most issues can be reduced to calculations on right angle triangles. Therefore, the majority of applications are related to right angle triangles. One exception to this fact is Spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is a part of hyperbolic geometry.
Trigonometry basics are frequently taught in school either as a separate course or as part of a precocious course.
Image Caption: A simple trigonometric triangle designed to show the parts of a right triangle. This triangle is also a 30-60-90 triangle. Credit: TheOtherJesse/Wikipedia