Thursday, April 8, 2010

A curious trigonometry limit

It is comforting to know that I still remember how to do high-school level pre-calculus. I was intrigued by the question "how to calculate the area of a regular convex polygon if we know the circumradius?" It turns out to be an easy exercise. Then I remembered that when the number of sides of a regular convex polygon increases to infinity, it becomes a circle. I decided to investigate this "limit" and found an interesting derivation.
\[ \lim_{x→0} {\sin{cx} \over x} = c \] which is a generalization of a well know trigonometry limit: \[ \lim_{x→0} {\sin{x} \over x} = 1 \]
The complete proof can be found in a short paper titled "A Curious Trigonometry Limit." (I'm also just beginning to learn how to use PGF and TikZ in LaTeX).
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