# 1. Angles

by M. Bourne

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).

### Examples

### Gradians

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.

Anti-clockwise, positive angle.

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

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 ° ' ".

## Exercises

Convert the following:

1) 36°23'47" to decimal degrees

2) 58.39° to DMS

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