In mathematics, one of six functions (sine, cosine, tangent, cotangent, secant, and cosecant) that represent ratios of sides of right triangles. They are also known as the circular functions, since their values can be defined as ratios of the x and y coordinates (seecoordinate system) of points on a circle of radius 1 that correspond to angles in standard positions. Such values have been tabulated and programmed into scientific calculators and computers. This allows trigonometry to be easily applied to surveying, engineering, and navigation problems in which one of a right triangle’s acute angles and the length of a side are known and the lengths of the other sides are to be found. The fundamental trigonometric identity is sin2? + cos2? = 1, in which ? is an angle. Certain intrinsic qualities of the trigonometric functions make them useful in mathematical analysis. In particular, their derivatives form patterns useful for solving differential equations.