Thursday, 15 May 2014

trigonometry - Why $x



In proof of limx0sinxx=1 is assumed that sinxxtanx while $0

Answer



It's fine if the comparisons of those lengths is intuitively clear to you, but if you want to be rigorous, it's easier to compare nested areas than it is to compare curve lengths.



Working off of David Mitra's picture (see his answer), the area of the triangle spanned by the lines cos(t) and sin(t) is 12cos(t)sin(t). That area rests inside the sector of angle t, which has area t2 (the proportion t2π of the entire circle's area π). And in turn, the area of sector is inside the triangle spanned by the horizontal radius of the unit circle and tan(t), which has area 12tan(t).




Thus we have the inequality 12cos(t)sin(t)t212tan(t)

Or cos(t)sin(t)ttan(t)
canceling the 2's. You should be able to adjust your proof of limt0sin(t)t=1 to utilize this slightly weaker inequality.


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