Trigonometry: Angles: Measuring Angles

There are three units of measure for angles: revolutions, degrees, and radians. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units.

Revolutions

A revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side. In trigonometry, angles can have a measure of many revolutions--there is no limit to the magnitude of a given angle. A revolution can be abbreviated "rev".

Degrees

A more common way to measure angles is in degrees. There are 360 degrees in one revolution. Degrees can be subdivided, too. One degree is equal to 60 minutes, and one minute is equal to 60 seconds. Therefore, an angle whose measure is one second has a measure of degrees. When perpendicularity is discussed, it is most often defined as a situation in which a 90 degree angle exists. Often degrees are used to describe certain triangles, like 30-60-90 and 45-45-90 triangles. As previously mentioned, however, in most cases that concern trigonometry, radians are the most useful and manageable unit of measure. Degrees are symbolized with a small superscript circle after the number (measure). 360 degrees is symbolized 360^o.

Radian

A radian is not a unit of measure that is arbitrarily defined, like a degree. Its definition is geometrical. One radian (1 rad) is the measure of the central angle (an angle whose vertex is the center of a circle) that intercepts an arc whose length is equal to the radius of the circle. The measure of such an angle is always the same, regardless of the radius of the circle. It is a naturally occurring unit of measure, just like Π is the natural ratio of the circumference of a circle and the diameter. If an angle of one radian intercepts an arc of length r, then a central angle of 2Π radians would intercept an arc of length 2Πr, which is the circumference of the circle. Such a central angle has a measure of one revolution. Therefore, 1 rev = 360^o = 2Π rad. Also, 1 rad = ()^o = rev.

Conversion between Revolutions, Degrees, and Radians

Below is a chart with angle measures of common angles in revolutions, degrees, and radians. Any angle can be converted from one set of units to another using the definition of the units, but it will save time to memorize a few simple conversions. It is particularly important to be able to convert between degrees and radians.