I’d ask what is so freaking useful about radians, but I’m afraid someone will tell me. Likely several someones. And then, though I use a calculator, I’d ask why anybody would use decimal degrees because 360 degrees in a circle is so elegant, in an aesthetic if not arithmetical manner, and the whole thread would go down the toilet with me looking like a clueless turd and I already have a wife to do that for me so I won’t.
The simplest answer is that in radians, sin(x) ~= x ~= tan(x), for small values of x. You also get the convenient result that the derivative of sine is cosine, without some awkward factor stuck on.
Or, they’ll just point you toward this explanation at BetterExplained.com (which also addresses the question of why there are 360 degrees in a circle).
Chronos and Thudlow Boink, I’m going to guess here that dropzone was right the first time — he probably didn’t want to know. :eek:
It could be worse. Someone could also explain why e is a better base for logarithms and exponentials than any other number. 
That last part is why my college calculus classes were less than a resonding success grade wise.
nm.
Oddly enough, it’s basically the same reason: It makes the slope at zero equal to 1, and it makes the derivative simple.
Oh, sweet Jesus, don’t somebody start on e! I was going through the manual of one of my many calculators the other day, saw what e equals in regular numbers, and asked what anybody would do with it. Asked rhetorically because Senegoid is right that I don’t want to know. Except I do. Or not.
You may ask why a mathematical semi-literate reads and comments on math threads and owns a bunch of powerful calculators when all he needs and knows how to use are four functions plus memory and square root (my Trig was never strong and seeped out of my brain when I stopped using it), still can’t do Algebra to save his life, but keeps trying to figure all of that out while he eats lunch. Good question; wish I could answer it. Math has always beaten me and before I die I’d like to beat it some.
thanks for great cite/site. “Don’t learn by rote.”
Fun stuff…
Introduction to compound interest and e (Khan Academy)
An Intuitive Guide To Exponential Functions & e (Better Explained)
And to put the same thing another way: Accurately plot 2[sup]x[/sup] and measure the gradient anywhere, and it comes out less than 2[sup]x[/sup]. Repeat for 3[sup]x[/sup], and this time the gradient’s always more. So somewhere in between there is a number e so that e[sup]x[/sup]'s gradient is always equal to e[sup]x[/sup].
Plus which, when you’re using radians and e, there’s a simple equation unifying e, i, p, 0 and 1.
Would you like to know if 0.999… = 1 (and if so, why)? 
Or on what day the blue-eyed islanders leave the island and why?
No one’s worked this joke in yet? I’m disappointed in you lot.
Who did the mathematician wink at? Acute angle.
Who did he marry, though? The right angle.
I have a theory, and I can’t remember if I read this somewhere or I made it up independently.
As the Earth goes round the Sun in (very) approximately 360 days, that was the thinking behind choosing 360 degrees (rather than 720 or 540 or 11 or whatever). Its multi-divisibility helped confirm it as extremely practical, and maybe even as Divinely inspired.
360 = 6 x 60.
In other words, it makes perfect sense in the Sexagesimal (base 60) system that Sumeria used 5000 years ago and which was later adopted by those Johnny-come-lately Babylonians.
Indeed. 90 degrees is one quarter of a turn (tau) around the circle, so, τ/4 makes more sense. See here.
Such a group is as real as the Illuminati.
Why do we all use base ten in everyday life? Same reason. We could all switch to base 8 or base 12 or base 16 if we wanted. No international standards body decides on that, either, by the way.
I’ve entertained my calc students in the past by computing exponentials for small values of x in my head. They tend to be impressed until they start picking out the easy trick.