1. Angles

by M. Bourne

Angles showing vertex, initial and terminal sides

An angle is a measure of the amount of rotation between two line segments. The 2 line segments (or rays) are named the initial side and terminal side as shown in the diagram.

If the rotation is anti-clockwise, the angle is positive. Clockwise rotation gives a negative angle (by convention).



There is another unit for measuring angles, called gradians. In this system, the right angle is divided into 100 gradians. Gradians are used by surveyors, but not commonly used in mathematics. However, you will see a "grad" mode on most calculators.

70 degree angle
Anti-clockwise, positive angle.
angle measured clockwise
Clockwise, negative angle.

Angles are commonly measured in degrees or radians. If you can't wait to learn about radians, see section 7. Radians.

Continues below

Standard Position of an Angle

Standard position of an angle on the cartesian plane

An angle is in standard position if the initial side is the positive x-axis and the vertex is at the origin. The 2 examples given above are in standard position.

We will use `r`, the length of the hypotenuse, and the lengths x and y when defining the trigonometric ratios in the next section.

Degrees, Minutes and Seconds

The Babylonians (who lived in modern day Iraq from 5000 BC to 500 BC) used a base `60` system of numbers. From them we get the division of time, latitude & longitude and angles in multiples of `60`.

Similar to the way hours, minutes and seconds are divided, the degree is divided into 60 minutes (') and a minute is divided into 60 seconds ("). We can write this form as: DMS or ° ' ".


Convert the following:

1) 36°23'47" to decimal degrees

2) 58.39° to DMS