# 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).

### 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.

### Examples

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

## 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 three examples given above are in standard position if the vertex is at (0, 0).

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

*x-*axis

*y-*axis

## 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 ^{o} ' ".

## Exercises

Convert the following:

1) 36^{o}23'47" to decimal degrees

Answer

Your calculator may be able to do this conversion for you directly. The question is similar to asking "How many hours in `36` hours, `23` minutes and `47` seconds?"

What is happening is:

36

^{o}= 36^{o}(we don't need to do anything to the whole number of degrees)23' = `23/60` of 1

^{o}= 0.38333^{o}47" = `47/3600` of 1

^{o}= 0.0130555^{o}

Adding them up, we get:

36^{o} + 0.38333^{o} + 0.0130555^{o} = 36.396386^{o}

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2) 58.39^{o} to DMS

Answer

We need to convert this to degree-minutes-seconds. Once again, your calculator may be able to do this directly. As always, it is good to know what the calculator is doing for you.

58

^{o}= 58^{o}(nothing to do here)0.39 of 1

^{o}= 0.39 × 60' = 23.4'. This means `23` minutes and `0.4` of a minute left over. We still have a decimal portion, so we need to find `0.4` of `1` minute.0.4 of 1' = 0.4 of 60" = 24".

Putting this together, we have 58.39^{o} = 58^{o}23'24".

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