Information AboutAngle |
| CATEGORIES ABOUT ANGLE | |
| elementary geometry | |
| trigonometry | |
| angleelementary geometry | |
| trigonometry | |
| angle | |
| elementary geometry | |
| physical quantity | |
|
In Geometry and Trigonometry , an angle (in full, '''plane angle''') is the figure formed by two Rays sharing a common Endpoint , called the Vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below).
HISTORY 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 used 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. MEASURING ANGLES In order to measure an angle θ, a Circular Arc centered at the vertex of the angle is drawn, e.g. with a pair of Compasses . The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen): : The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio ''s''/''r'' is unaltered. In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a Spiral using Polar Coordinates , an extra full turn gives rise to a quite different point on the curve. Units Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to ''n'' units, for some whole number ''n'' (for example, in the case of degrees, ''n'' = 360). This is equivalent to setting ''k'' = ''n''/2''π'' in the formula above. (To see why, note that one full circle corresponds to an arc equal in length to the circle's Circumference , which is 2''πr'', so ''s'' = 2''πr''. Substituting, we get ''θ'' = ''ks''/''r'' = 2''πk''. But if one complete circle is to have a numerical angular value of ''n'', then we need ''θ'' = ''n''. This is achieved by setting ''k'' = ''n''/2''π''.)
Positive and negative angles A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured Counterclockwise , and '''negative angles''' if measured Clockwise , from a given line. If no line is specified, it can be assumed to be the X-axis in the Cartesian Plane . In many geometrical situations a negative angle of −''θ'' is effectively equivalent to a positive angle of "one full rotation less ''θ''". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to a counterclockwise rotation of 360° − 45° (that is, an angle of 315°). In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a Vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In Navigation , Bearings are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees. Approximations
TYPES OF ANGLE |
|
|