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Radians and Degrees (page 1 of 2) Sections: Radians and Degrees, Reducing Angles to Useful Values Radians and degrees are two units for measuring angles. There are at least four such units, but degrees and radians are the ones you are most likely to encounter in high school and college. Degrees Degrees are used to express directionality and angle size. If you stand facing directly north, you are facing the direction of zero degrees, written as 0°. If you turn yourself fully around, so you end up facing north again, you have "turned through" 360°; that is, one revolution (one circle) is 360°. Aside: Why is one revolution divided into 360 parts called "degrees"? Because the ancient Babylonians, now four or fivethousand years dead, viewed the numbers 6, 12, and 60 as having particular religious significance. It is because of them that we have twelvehour nights and twelvehour days, with each hour divided into sixty minutes and each minute divided into sixty seconds. And "once around" is divided into 6×60 = 360 parts called "degrees". So a full revolution is 360° and a halfturn (an "about face") is 180°. If you start by facing north again and then turn to the east, you will have made a 90°, or onequarter, turn, and you will be facing toward 90°. If you start facing north and then turn to the west, you will have made a 90° turn, but you will be facing toward 270°. This is because directional degrees start at 0° for "north" and then go around clockwise.
If, when making your onequarter turn from "north" to "west", you held your arm straight out in front of you, your arm would be said to have "swept out" a 90° angle. This angle would have been formed by your arm's starting position (the "initial" side of the angle) and your arm's ending position (the "terminal" side of the angle). The path of your fingertips as your arm moved would be an "arc" and the angle you turned through is said to "subtend" that arc. Note: When directions are given in terms
of degrees, the direction is found by starting at "north", being
0°, and moving clockwise by the number of degrees given. Another way of
giving directions using degress is of the form N36°W or S27°E. These mean
"36 degrees west of north" and "27 degrees east of south",
respectively. Whatever convention your book uses should be defined in
the book; ask your instructor, if it isn't otherwise clear. And yes, this is different from how you'll be measuring angles. Then you'll start with 0° being "east", sort of, and you'll rotate anticlockwise. Copyright © Elizabeth Stapel 20102011 All Rights Reserved Decimal Degrees and DMS When you work with degrees, you'll almost always be working with decimal degrees; that is, with degrees expressed as decimal numbers such as 43.1025°. But just as "1.75" hours can be expressed as "1 hour and 45 minutes", so also "degrees" can be expressed in terms of smaller units. These units, just as for "hours", are called "minutes" and "seconds". Just as "hours" can be expressed as decimals or else as hours  minutes  seconds, so also "degrees" can be expressed as decimals or else as degrees  minutes  seconds, denoted as "DMS".
I can see that I have 43°, but what do I do with the "0.1025" part? I treat it like a percentage of the sixty minutes in one degree, and find out how many minutes this is:
...or 6 minutes and 0.15 of a minute. Each minute has sixty seconds, so:
Then 43.1025° = 43° 6' 9" Notice the symbols: A single quotemark (an apostrophe) indicates "minutes" and a double quotemark indicates "seconds". This is similar to the notation (in Imperial measurements) for "feet" and "inches": the smaller unit gets the moresubstantial mark.
Clearly, I've got 102°, but how do I convert the minutes and seconds to decimal form? By using the definitions and doing the divisions. The 45' means 45/60 of a degree, since each degree contains sixty minutes. Simplification and long division gives me 45/60 = 3/4 = 0.75. So the 45' is 0.75°. Now I need to deal with the 54". Since each minute is sixty seconds, then I get 54/60 = 9/10 = 0.9. But this is minutes. Now I need to convert the 0.9 of a minute to degrees: (0.9 minutes)(1 degree / 60 minutes) = 0.015 degrees So 102° 45' 54" = 102° + 0.75° + 0.015° = 102.765°. Radians Why do we have to learn radians, when we already have perfectly good degrees? Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc). The 360° for one revolution ("once around") is messy enough. Why is the value for one revolution in radians the irrational value 2π? Because this value makes the math work out right. You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to make sense, the "numerical" value corresponding to 360° needed to be defined as (that is, needed to be invented having the property of) "2π is the numerical value of 'once around'." Converting Between Radians and Degrees Each of radians and degrees has its place. If you're describing directions to me, I'd really rather you said, "Turn sixty degrees to the right when you pass the orange mailbox", rather than, "Turn onethird π radians" at that point. but if I need to find the area of a sector of a circle, I'd rather you gave me the numerical radian measure that I can plug directly into the formula, rather than the degree measure that I'd have to convert first. But you won't always be given angle measures in the form you'd prefer, so you'll need to be able to convert between radians and degrees. To do this, you'll use the fact that 360° is "once around", and so is 2π. However, you'll use this fact in the form of the somewhat simplified correspondence of 180° to π.
Since 180° equates to π, then: The equivalent angle is
The equivalent angle is 30° Note that the way I used the correspondence varied with what I was given. If I needed to end up with radians, I put π on top; if I needed to end up with degrees, I put 180° on top. That's all there is to that. Top  1  2  Return to Index Next >>





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