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.
Examples
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.
(There is another unit for measuring angles, called gradians. 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.)
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
- 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° = 36° (we don't need to do anything to the whole number of degrees)
23' = 23/60 of 1° = 0.38333°
47" = 47/3600 of 1° = 0.0130555°
Adding them up, we get:
36° + 0.38333° + 0.0130555° = 36.396386°
2) 58.39° 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° = 58° (nothing to do here)
0.39 of 1° = 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° = 58°23'24".
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