Trigonometric functions

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. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis. The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed. In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length. Given an acute angle A of a right-angled triangle (see figure) the hypotenuse h is the side that connects the two acute angles. The side b adjacent to A is the side of the triangle that connects A to the right angle. The third side a is said opposite to A. If the angle A is given, then all sides of the right-angled triangle are well defined up to a scaling factor. This means that the ratio of any two side lengths depends only on A. These six ratios define thus six functions of A, which are the trigonometric functions. More precisely, the six trigonometric functions are: In a right angled triangle, the sum of the two acute angles is a right angle, that is 90° or π 2 { extstyle {frac {pi }{2}}} radians. This induces relationships between trigonometric functions that are summarized in the following table, where the angle is denoted by θ {displaystyle heta } instead of A. In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees.