The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional.
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