Trigonometry: Angles

Angles Defined

An angle is the union of two rays that share a common endpoint. The rays are called the sides of the angle, and the common endpoint is the vertex of the angle. The measure of an angle is the measure of the space between the rays. It is the direction of the rays relative to one another that determine the measure of an angle.

In trigonometry, angles are often defined in terms of rotation. Consider one ray, and then let it rotate a fixed distance about its endpoint. The ray in its initial position before the rotation, and the ray in its ending, or terminal position, after the rotation, creates an angle. The endpoint point about which the ray rotates is the vertex. The amount of rotation determines the measure of the angle. The ray in the initial position, before the rotation, is called the initial side of the angle. The ray in the terminal position, after the rotation, is called the terminal side of the angle. An angle created this way has a positive measure if the rotation was counterclockwise, and a negative measure if the rotation was clockwise.

Note that a ray can rotate all the way around to its initial position, and possibly further, and still result in an angle with an initial and terminal side. This definition of an angle places no restriction on the magnitude of an angle (how far it can rotate).