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An angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ἄγκοσ, a bend; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry.
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was utilized by Eudemus , who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch , who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.
A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwiseClockwise can refer to: Clockwise and counterclockwise Clockwise (movie)., from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane.
In navigation and other areas this convention may not be followed.
Mathematicians generally prefer angle measurements in radians because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios ofs can be developed into particularly simple Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point if their arguments are specified in radians.