![]() ![]() However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics). For this purpose, any angular unit is convenient. In geometric applications, the argument of a trigonometric function is generally the measure of an angle. Parentheses are still often omitted to reduce clutter, but are sometimes necessary for example the expression sin x + y Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin( x). Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.Ĭonventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. ![]() ![]() To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used then the domain of the other functions is the real line with some isolated points removed. 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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