Angles and Angle Measure

A geometric figure consisting of two rays with a common endpoint is called an angle. The common endpoint is called the vertex of the angle, and the rays are called the sides of the angle.

In the study of trigonometry, it is useful to generalize the concept of angle by letting the vertex coincide with the origin of the $xy$ plane, and begin by letting both sides coincide with the positive $x$ -axis. One side of the angle, called the initial side, remains fixed on the positive $x$ -axis. The other side of the angle, called the terminal side, is then ``rotated'' until it reaches its terminal position. Such an angle is said to be in standard position.

A positive angle corresponds to a counterclockwise rotation, while a negative angle corresponds to a clockwise rotation. Generalizing the angle concept in this manner, we can consider angles of any size, positive or negative, since the rotation of the terminal side can exceed a complete revolution (i.e. exceed $360^\circ,$ ) and in either direction. In short, an angle in standard position may be considered with any desired real number measure.

Two angles in standard position in the same coordinate system are called coterminal angles if their terminal sides coincide. An angle in standard position with terminal side coinciding with a coordinate axis, is called a quadrantal angle.

A central angle of a circle is an angle with a vertex that coincides with the center of the circle, and sides that correspond to radii of the circle. The arc formed by the intersection of these sides and the circle itself, is said to subtend the angle, and the angle is said to be subtended by the arc.