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A few days ago I was seeing one of those beautiful result videos. It was the integral of sin(x)/x from 0 to infinity.the whole integral finally boils down to the inverse tangent of  x as x approaches infinity and they obtain the result pi/2. They call it beautiful but I dont think that is the case because the pi/2 is obtained only as a result of our definition of radians. Had it been the inverse tangent in degrees, we would have gotten 90 which wouldn't be so impressive. I see this with a lot of other results that finally boil down to some trig function yielding a multiple of pi and dont find it so beautiful. Either im missing something that I shouldn't or that these "beautiful" results arent so "beautiful". Any thoughts?
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I think the point of the “beautiful” label is that perhaps it’s not obvious from the outset that we’re going to get a nice multiple of previously understood constants.  The fact that the area under the whole curve is pi/2 is not clear before carrying out the mathematics.  For example, try finding the area under something like arcsinx/sinx.  This appears to just be some random number, about 2.31883.  While there is nothing less numbery about that number, I would certainly say it is less “beautiful,” unless we were somehow able to figure out a closed form in terms of known constants.
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Consider a calculus formula like d/dx(sinx) = cos(x). The key thing to notice isthat this is only true when x is in radians. To get a similair formula for degrees, you would write:  
d/dx(sin(x degrees)) = d/dx(sin(pi\*x/180 radians)) = pi/180 cos(x degrees). You could choose any other angle measure, and you would get a similair constant at the front. The "beautiful" thing about radians is that constant is exactly 1.  


During the derivation of your problem, they will have used the fact that the integral of 1/(1+x\^2) is arctan(x), but similairly, this is only true when x is in radians. If you wanted to use arctan(x) in degrees, you woud get a constant of pi/180 in front that would cancel out the 90 degrees, giving pi/2 again.
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lim ₓ→₀ (sin x/x) = 1, where x is radians.
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