Angle in Geometry
An angle in geometry is a figure formed by two rays having a common vertex (origin). The rays are called sides of the angle. Consider a line extending indefinitely in both directions. A point O on the line divides it into two parts, one on each side of the point. The parts of the line on one side of the point O is called a ray or half-line. Thus, O divides the line into two rays OA and OA’.
The point O is called vertex or the origin of each of these rays. The word ‘ray’ is suggested by rays emanating from a source O.
An angle is denoted by the symbol $ \displaystyle \angle $, followed by three capital letters of which the middle one denotes the vertex of the angle and two other points are the points on the two sides or simply capital letters denoting the vertex.
Example: $ \displaystyle \angle AOB,\text{ }\angle AOC$ etc.
With each angle a number is associated, this number is called measure of the angle.
In geometry an angle always lies between $ \displaystyle {{0}^{\circ }}$ and $ \displaystyle {{360}^{\circ }}$ and negative angle has no meaning. Measure of an angle is taken to be the smallest amount of rotation from the direction of one ray of the angle to the direction of the other.
Angle in Trigonometry
In trigonometry the idea of angle is more general and it may be positive or negative and of any magnitude.
As in case of geometry, in trigonometry also the measure of angle is the amount of rotation from the direction of one ray of the angle to the other. The initial and final positions of the revolving ray are respectively called the initial side (arm) and terminal side (arm) and the revolving line is called the generating line or the radius vector. For example, if OA and OB be the initial and final positions of the revolving ray then angle formed will be $ \displaystyle \angle AOB$.
Angles exceeding $ \displaystyle {{360}^{\circ }}$
In geometry we confine ourselves to the angles from $ \displaystyle {{0}^{\circ }}$ to $ \displaystyle {{360}^{\circ }}$, but there may be problems in which rotation involves more than one revolution, for example, the rotation of a flywheel. In trigonometry we generalize the concept of angle to angles greater than $ \displaystyle {{360}^{\circ }}$, this angle can be formed in the following ways:
The revolving line (radius vector) starts from the initial position $ \displaystyle \overrightarrow{{OA}}$ and and makes n complete revolutions in anticlockwise direction and also a further angle $ \displaystyle \alpha $ in the same direction. We thenhave a certain angle $ \displaystyle {{\beta }_{n}}$ given by
$ \displaystyle {{\beta }_{n}}={{360}^{\circ }}\times n+\alpha $
Where $ \displaystyle {{0}^{\circ }}\le \alpha \le {{360}^{\circ }}$ and n is a positive integer or zero.
Thus there are infinitely many angles $ \displaystyle {{\beta }_{n}}$ with initial side $ \displaystyle \overrightarrow{{OA}}$ and final side $ \displaystyle \overrightarrow{{OB}}$.
For example $ \displaystyle {{\beta }_{o}}=\alpha ,\text{ }{{\beta }_{1}}={{360}^{\circ }}+\alpha ,\text{ }{{\beta }_{2}}={{720}^{\circ }}+\alpha $ etc.
Sign of Angles
Angles formed by anticlockwise rotation of the radius vector are taken as positive where as angles formed by clockwise rotation of the radius vector are taken as negative.
