# Trigonometry

**Trigonometry**, branch of geometry that deals with the ratios of the sides and angles of triangles, particularly of right-angled triangles and the applications of these ratios. Plane trigonometry deals with these relationships mapped on a plane surface. The principle ratios, when considering angle A of right-triangle ABC whose sides opposite the angles A, B, and C respectively are a, b, and c, and where c is the hypotenuse, are:

name | abbreviation | ratio |

tangent | tan A | a/b |

sine | sin A | a/c |

cosine | cos A | b/c |

cotangent | cot A | b/a |

cosecant | cosec A | c/a |

secant | sec A | c/b |

As can be seen, the cotangent is the reciprocal of the tangent, the cosecant that of the sine, and the secant that of the cosine. The basis of trigonometric calculations is Pythagoras's theorem, a^{2} + b^{2} = c^{2}, which in trigonometric form reads sin A^{2} +cos A = 1; this is true for angle A. From these ratios are derived the trigonometric functions, setting y equal to tan x, sin x, etc. These functions are termed transcendental (nonalgebraic). Of particular importance is the sine wave, in terms of which many naturally occurring wave motions, such as sound and light, are studied. Spherical trigonometry studies triangles lying on a sphere, and thus can be used to calculate distances on a globe.

## Additional topics

21st Century Webster's Family Encyclopedia21st Century Webster's Family Encyclopedia - Transcendentalism to United Church of Christ