All of the programming languages I use think in radians, as do the spreadsheets, and only Excel has a command to convert to degrees, as used by God, the angels, and most of humanity. Why radians? It’s weird enough dividing half a circle into 180 degrees but why on earth would you want to divide it into an irrational number (pi) of units? Is this just some archaic leftover from the days of slide rules like logarithms, when folks had to make do with bizarre tricks because they didn’t have calculators, or is there a valid mathematical reason for using them?
Radians are an accurate measure of arc length. You want to know how long the arc is in a quarter of a circle? it’s radius * pi/2., and there’s pi/2 radians in a right angle. Similarly, the length of the whole arc of the circle? Also called circumference: radius * 2*pi.
All kinds of formulae use radians as a natural number. Otherwise you’d have to be dividing it into all kinds of ratios to do the math. Of course, that’s only a practical use.
In the real world, a radian is “natural” in the sense that it occurs all but itself. A circle has a circumference of 2pi radians. But what’s 360 degrees? That’s some made up, non-natural number. Or my metric degrees – my circles have 100 metric degrees. Start applying that to real world problems. And have you ever done degrees-minutes-seconds just for the fun of it?
One of the reasons is calculus. The derivative of sin(x) is cos(x), but only if you measure in radians; if you measure in degrees, then the derivative of sin(x) is pi*cos(x)/180. There’s also geometry: the length of an circular arc is the product of the radius and the angle, and the area of a circular sector is half the square of the radius times the angle, but again only if the angle is measured in radians. For both reasons, radians are a very natural unit of measurement for angles.
Keep going. I’m not yet convinced they exist solely to make using a slide rule faster.
?
Slide rules have nothing to do with it. It’s the natural measurment of an angle. Degrees are totally contrived and non-sensical. The benefit of a degree is our favorite angles (180, 90, 45, 60, 30, etc) are nice whole numbers. This makes describing angles (trivially) easier, but doesn’t help for any sort of meaningful math. If you need to figure out area of an wedge, length of an arc, etc, and you have the angle in degrees, the first step will be to convert it to radians, anyway.
The area of a wedge of radius “r” and with a sweep of 1 radian is r. Radians are, quite simply, the natural unit of angle.
Also, as Orbifold points out, trigonometric functions expressed in radians have desirable analytic properties; in addition to the ones noted above, there’s the fact that sin(t)/t approaches 1 as t approaches 0; this is only true when the sine function is defined in radians.
As noted above, radians are a “natural” measure of angle. Dividing the length of the arc subtended by an angle by the radius of the circle is the angle in radian measure. It’s entirely independent of how many fingers we have (although, if you express the result in decimal numbers, of course, your answer will certainly be related to the number of fingers we have.) or our astronomical setup, or anything. In just the same way, the number ]e is the “natural” base for logarithms, which could take any base.
The basis for degree measure has to be the length of the year (365 and about a quarter days), modified slightly to produce 360, a number evenly divisible by a lot of other numbers. The near-coincidence is too close, for me, to otherwise explained. And if the Babylonians wanted a number that was evenly divisible by a lot of other factors, there were a lot of other coices.
Grads, though, are an offense against nature and common sense.
[QUOTE=aktep]
?QUOTE]I don’t use slide rules and I don’t use radians. The relationship is obvious. 
I’ve never even seen a slide rule, and I used to use radians all the time
Perfectly logical.
Finally, we agree on something.
You can think of an angle measurement in radians as being simply the ratio of the subtended circumference to the radius. That’s a dimensionless number, making many calculations much easier. You can also use radians as dimensional, equal to 57+ degrees, if *that * will make your calculation easier.
I think Elvis nailed my problem with radians. To me an angle is a dimension or a size, something visualized and measured in the real world. To the rest of you it is a number and need not have any physical existence.
However, this discussion has been helpful. Since I never think things through too deeply I always assumed the charts and formulae I’ve used that used the number .01745 used it because that was the sine of one degree rounded off to five decimal places. Now I suspect they use it because .01745 is one degree converted to radians and rounded off to five decimal places.
But the relationship between angle-as-radians and arc length is a “real world” thing.
Yeah? Do you have a protractor with radians I can give to the guys in the shop?
The reason why this works, by the way, is because sin(t) is very nearly t for small t – but only when t is expressed in radians. Which is another reason why radians are cool.
In my previous post I misstated the formula for the area of a wedge (it’s r<sup>2</sup>/2) because I confused it with the formula for the length of a radius r arc of one radian (which is r). I beg indulgence for this oversight.
I’m willing to bet that, whatever formulas or procedures the computer uses to compute sines, cosines, arctangents, etc., it works with radians. To give you the answer in degrees (or to accept degrees as an argument) would involve the same procedure plus an additional step to convert.
As Orbifold noted, there are things in Calculus that only work (or at least only work nicely) if you think in radians, not degrees. One more example: as long as x is not too big, sin x is approximately equal to x - x[sup]3[/sup]/6 + x[sup]5[/sup]/120, but only if x refers to radians. (Those of you who’ve studied Taylor polynomials and/or Taylor series, probably in Calculus II, will know where this comes from.)
But not inherent in the circle itself. The year doesn’t have to be that long. However, a circle of radius 1 must have circumference 2\pi (in the Euclidean plane).
More generally, consider a sector of radius r and angle heta. What’s the length of the curved side? r* heta. What’s its area? r[sup]2[/sup]* heta/2. Simple.
I agree, grad students smell funny.
Anyway, dropzone, a protractor marked in radians would be marked with: pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, pi. It’s how I’ve visualized angles for years now.
And that, in case you missed it, was the point of my contribution.