Fundamental Concepts

Angles

An angle is formed by rotating a ray around its endpoint. That endpoint is called the vertex of the angle. The starting position of the ray is called the initial side, and the ending position after rotating the ray is called the terminal side. The position in which the initial side is on the x-axis, and the vertex is at the origin is the standard position. Positive angles are rotate counterclockwise, and negative angles rotate clockwise. Angles that share a common terminal side are called coterminal angles.

Radians

One way to measure angles is in radians. This method of measure is mostly used in trigonometry and calculus.
Definition: One radian is the measure of the angle that intercepts an arc i.e. of a circle. That arc length must be equal in length to the radius r of the circle. Because the circumference of a circle is 2r, the angle measure of one full revolution counterclockwise corresponds to an arc length of 2r. So 2 radians is the same as 360°, radians is 180°, and /2 radians equals 90°.
As 2 is approximately 6.28 there are just over six radius lengths in a full circle. In general, the radian measure of a central angle ? is obtained by dividing the arc length s by the radius r.
Recall that the four quadrants in a coordinate system are numbered I, II, III, and IV. Note that angles between 0 and /2 are acute and that angles between /2 and are obtuse.
Two angles are coterminal if they have the same terminal side. For instance, the angles 0 and 2 are coterminal, as are the angles /2 and 5/2. You can find an angle that is coterminal to a given angle by adding or sub-tracting 2 (one revolution). A given angle has infinitely many coterminal angles. For instance, is coterminal with + 2n where n is any integer.

Degrees

A second way to measure angles is in terms of degrees. A measure of one degree (1°) is the same as a rotation of 1/360 of a full revolution about the vertex of the angle. So, a full revolution (counterclockwise) corresponds to 360°, a half revolution is 180°, a quarter revolution is 90°, and so on. Because 2 radians corresponds to one complete revolution, degrees and radians are related by the equations:

360° = 2 radians and 180° = radians

From that, you obtain

1° = /180 radians and 1 radian = 180°/

Conversion Rules:

To convert degrees to radians, multiply degrees by /180°

To convert radians to degrees, multiply radians by 180°/

When no units of angle measure are specified, radian measure is implied.