Naormally just the first term is used and referred to as the “small angle assumption”. Very rarely do people use the second and third terms without just calculating the full (or as full as your calculator will do) sine.
Bloody hell. My post obviously refers to what Thudlow Boink wrote. Simulposts, ech.
HEY! :mad:
Maybe you can afford all kinds of fancy perfumes and colognes, but they don’t pay people that well to know why radians are the One True Angle Measure.
Psssst! Cal! In case you missed it, and you did, Mathochist was responding to my tiny response to your larger statement. But let’s stop dumping on each other and go back to dumping on mathophobic me.
Yes that is the case in all the software I have examined. Usually a Taylor series is used or occasionally one of the faster but more limited approximations. In any case, the result naturally drops out in radians. It is sensible to use radians in computer software if only to avoid conversion to and from a quite arbitary scale such as 360 degrees.
No worries, man. After next year I’ll be a physics grad student, which as we all know are the worst of the lot.
OTOH, I’m paid rather well to express angles in degrees. So who’s the smart guy? 
The number 60 (or 6*60)has some very nice properties, being evenly divisible by the first three prime numbers. The Babylonians probably appreciated this due to their (apparent) lack of knowledge of fractional math. It does require that you remember an extraordinary number of symbols, though. These days, hexadecimal math would be a lot more convenient, but first we’d have to figure out how to genetically modify human children to have three extra fingers per hand.
I can’t disagree there, especially since people often confuse grads and % slope (not the same, but I’ve argued the point repeatedly). Degrees don’t really make any more sense, though, except to satisfy our desire to chop measurements up into more conceptually managable segements.
As for the OP, as other have said, radians fall out of the proofs of trigonometic functions and are a natural measurement of angle. One can see this with Euler’s Identity,
e[sup]*i*θ[/sup] = cos(θ) + *i*sin(θ)
which, when set to θ=π gives you
e[sup]*i*π[/sup] + 1 = 0
the most beautiful relationship in natural mathematics.
Radians also very handy when resolving polar vectors in linear circuits and/or mechanism design, among other things. It just doesn’t sound so impressive when the gunnery chief is yelling, “Elevate to 1/3 pi radians!”
Stranger
Yes, let’s 
Cal, there are a lot of posts going up very quickly, tensions are high, and it’s easy to take a miscue. Let’s remember who the real enemies are: innumerates.
Now… degrees to radians, dropzone, convert or die!
Easy! I multiply the degrees by .01745. Of course, it helps that I have several calculators in reach.
[aside]
Back in the early days of calculators my Trig prof made us use logs because “what are you going to do out in the field if your calculator’s batteries die?” He didn’t explain what we were to do if we didn’t have a log table handy; perhaps in his world log tables were as ubiquitous as calculators are in mine. And as this was eight years before I bought my Casio fx-82B that is still on its original batteries after twenty years the threat of batteries dying was real.
[/aside]
No need to memorize that number. Just divide by 180 and multiply by pi. In programming, define “deg=pi/180” once and every time you use a trig function, write e.g. “sin(x * deg)”. Either way, it’ll give you as many significant digits as you (or the compiler) know the digits of pi.
I’ve used it so often I’ve already memorized it, but as I noted above I now know why I had to memorize it.
Even if you never measure a single angle in your life or draw a single triangle, if you do enough calculus you’ll still eventually find a reason to define the function sin(x), because it’s one of the solutions to the differential equation y’’+y=0. And if you plot out that solution, you’ll find that it has a period of 2*pi…in other words, that it’s the version of sin(x) that uses radians, not degrees.
And if you think differential equations don’t get “visualized and measured in the real world”, then I’m afraid you’re beyond help
Speak for your stinky self! I smell of roses and sweetness and light. 
Oh, and the reason for radians? To confuse the heck out of already befuddled astronomers.
As I noted in this thread,
It turns out i made up a precuror of differential equations (approximating the area of an irregular shape by slicing it into lots of rectangles) on the job once so I know what you mean.
And, to complete the circle, percent slope is very close to the angle measured in radians (a 2% slope is very close to an angle of 0.02 radians).
Welcome to integral calculus.
Why not octal math – just ignore those strange looking thumbs. And not meaning to hijack (yes I do). Why did hexadecimal math get started – needing to introduce those extra numerals – rather than octal math in the first place. Was it because they wanted to work with four bits rather than 3 bits?
Probably. 3 isn’t a very nice number, and octal will generate “longer” numbers than decimal. 4 is a nice power of two, and I’m sure the digital electronics industry, laden with powers of 2, had a heavy influence here.
Hex numbers have some very nice properties:
10[sub]H[/sub] = 4[sup]2[/sup] = 2[sup]4[/sup]
meaning that squares and higher even powers come up more regularly in “hexades”, which is convenient (notationally) when dealing with squared or root quantites. I rather like the Hex Clock personally.
Octol offers less “units” per digit (i.e. nn only covers 0-63, nnn only covers 0-511) and groups bytes into a rather awkward three digit binary pattern. Hex is certainly more widely used; the only place I can think of offhand is the use of octal in specifing user permissions with chmod, though I’m sure some Unix guru can come up with much more. But hex is used all over the place to code instructions and represent data.
Stranger