Radians, Degrees, and Converting Between
Purplemath
Radians and degrees are two types of units for measuring angles. There are very many such units (such as "gradians" and "MRADs"), but degrees and radians are the ones you are most likely to encounter in high school and college.
Degrees
Degrees are used to express both directionality and angle size.
If you stand facing directly north, you are facing the direction of zero degrees, written as 0°. (The superscripted "circle" stands for "degrees".) If you turn yourself fully around, so you end up facing north again, you have "turned through" 360°; that is, one full revolution (or one circle) is 360°.
Why is one revolution divided into 360 parts called "degrees"? Because the ancient Babylonians, now four- or five-thousand years dead, viewed the numbers 6, 12, and 60 as having particular religious significance.
It is because of them that we have twelve-hour nights and twelve-hour days, with each hour divided into sixty minutes and each minute divided into sixty seconds. Also their fault is the fact that "once around" (that is, one full revolution) is divided into 6×60 = 360 parts called "degrees".
So a full revolution is 360° and a half-turn (or an "about face") is 180°. If you start by facing north and then turn to the south, you'll have made a half-turn, half of a revolution, or gone half-way around a circle. You'll also have "turned through" 180°.
If you start again by facing north and then turn to the east, you will have made a 90°, or one-quarter, turn, and you will be facing toward 90°. If you start facing north and then turn to the west, you will have made another 90° turn, but this time you will be facing toward 270°. This is because directional degrees (usually) start at 0° for "north" and then go around clockwise.
If, when making your one-quarter 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 (usually) 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 specifically defined in the book; ask your instructor, if it isn't otherwise clear.
And yes, this way of measuring direction (namely, starting at north and moving clockwise) is different from how you'll be measuring angles. When you're doing graphs and drawings involving measured angles, you'll be starting with 0° being "east" (it'll actually be the x-axis), and you'll rotate anti-clockwise.
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 decimal hours or else as "hours - minutes - seconds", so also "degrees" can be expressed as decimal degrees or else as "degrees - minutes - seconds", denoted as "DMS".
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Convert 43.1025° to DMS form.
I can see that I have 43°, but what do I do with the "0.1025" fractional part of a degree?
I will treat this fractional portion like a percentage of the sixty minutes in one degree. Using this reasoning, I can then find out how many minutes are in this percentage of a degree:
...or 6 minutes and 0.15 of another minute.
Each minute has sixty seconds. I can apply the same reasoning and method as I did for the fractional portion of a degree to this fractional portion of a minute:
Then 43.1025° is equal to 43 degrees, 6 minutes, and 9 seconds, or, in DMS notation:
43° 6' 9"
Notice the symbols that I used in my answer above. You already knew that the superscripted circle stood for "degrees". Now you can see that a single quote-mark (an apostrophe) indicates "minutes" and a double quote-mark indicates "seconds".
This is similar to the notation (in Imperial measurements) for "feet" and "inches". You can keep the notations straight by remembering that, just as is the case with "feet" and "inches" the smaller unit (namely, the "seconds") gets the larger marker (namely, the double quote-mark).
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Convert 102° 45' 54" to decimal form.
Clearly, I've got 102°, but how do I convert the minutes and seconds to decimal form?
I'll do the conversion by using the definitions of "degrees", "minutes", and "seconds"; and by doing the appropriate divisions.
Each degree contains sixty minutes. Then the 45' means that I have of a degree. Simplification of this fraction, and then doing the long division, gives me:
So the 45' is 0.75°. (This is similar to 45 minutes of time being 0.75 of an hour.)
Now I need to deal with the 54". Since each minute consists of sixty seconds, then I get:
But this number, 0.9, is in terms of minutes; it stands for "nine-tenths of one minute of arc". I need to convert the 0.9 of a minute to a value in terms of degrees. Since there are sixty minutes in one degree, then:
Adding these up, I get:
102° 45' 54"
= 102° + 0.75° + 0.015°
= 102.765°
Then 102° 45' 54", in decimal form, is equal to:
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' a circle."
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 (1/3)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 also is 2π. However, you'll use this equivalence fact in the form of the somewhat simplified correspondence of 180° to π.
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Convert 270° to radians.
I know that 180° equates to π, so I can use this relationship to do the conversion. I've got degrees and I want radians, so I'll want "degrees", as a unit, to cancel off. Since they gave me degrees, then "degrees" is currently on top (of a fraction, over "1"), so I'll put the "180" for "degree" underneath when I multiply, to get the cancellation I need.
Then the equivalent angle, in radians, is:
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Convert
radians to degrees.
I need to convert from radians to degrees, so I'll use my conversion factor with the "radians" on the bottom, so the unit that I don't want will cancel off:
Then the equivalent angle, in degrees, 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 this unit conversion.
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