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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).


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.


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 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.

initial side terminal side θ r x y (x, y) (0, 0)

An angle in standard position.

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 ' ".


Convert the following:

1) 36o23'47" to decimal degrees


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:

36o = 36o (we don't need to do anything to the whole number of degrees)

23' = `23/60` of 1o = 0.38333o

47" = `47/3600` of 1o = 0.0130555o

Adding them up, we get:

36o + 0.38333o + 0.0130555o = 36.396386o

2) 58.39o to DMS


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.

58o = 58o (nothing to do here)

0.39 of 1o = 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 58.39o = 58o23'24".


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