Specifying a direction is required in many situations, particularly in the nautical world. One of the ways to specify a direction is to state its bearing. More to the point, bearings are a way of expressing angles. Trigonometry is the branch of mathematics that deals with angles. In trigonometry, there are two primary ways of specifying a bearing, a true bearing and a conventional bearing. Converting from both kinds of bearings to an angle is simple and can make deciphering a direction easy.
Compass Direction

The four primary compass directions are north, west, east and south. Each corresponds to an angle. North corresponds to 0 degrees, east is 90 degrees, south is 180 degrees and west is 270 degrees. When it comes to specifying a bearing, north corresponds to the direction that is forward of the current position. In mathematics, north is usually specified as up.
True Bearings

A true bearing is a single number. It is specified as an angle from north. From the direction forward or up, rotate clockwise the specified number of degrees. For a line that is terminated by two points A and B, each point has a true bearing ascribed to it. By placing a vertical line at each point, and counting the degrees from the vertical line clockwise to the drawn line, the bearing of each point can be determined. For a straight line, one point's bearing will be 360 minus the other point's bearing.
Conventional Bearings

A conventional bearing gives three pieces of information: north or south, specifying from which direction the bearing starts; a number specifying the number of degrees; and either west or east, specifying the direction from north or south the direction is. For example, S30W means that the direction is 30 degrees clockwise from south. N40W means 40 degrees counterclockwise from north. For a conventional bearing, there are usually two interchangeable bearings, one for north and one for south. It is a simple case of changing north to south or vice versa, and replacing the angle with 180 minus the angle. In the previous example of S30W, the bearing could also be written as N150W.
Converting Bearings

To convert from a true bearing to a conventional bearing, first determine which of north or south the bearing is closer to. Then determine the number of degrees from that direction the bearing is placed. Finally note to which side, west or east, the bearing is. For example, a true bearing of 190 degrees is 10 degrees from south in the direction of west. The conventional bearing is then S10W. To go back, just count the degrees from north in the clockwise direction.
Conversion to a Mathematical Angle

One can also convert a bearing to a mathematical angle. A mathematical angle counts degrees counterclockwise from the positive x direction. This is the angle that is used in sine or cosine calculations. To covert to a mathematical angle, start with a true bearing. Then subtract it from 90. A true bearing of 50 degrees is a mathematical angle of 90  50 = 40 degrees. A true bearing of 105 degrees is an angle of 90  105 = 15 degrees.
References
 "Trigonometry"; Ron Larson and Robert P. Hostetler; 2006
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