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 May 21 comment How to derive the law of cosines without the pythagorean theorem I don't buy it. The Pythagorean Theorem is a special case of the law of cosines. I won't believe that there's not a proof of the latter (independent of the former) until I see a proof that it can't be done. May 21 asked How to derive the law of cosines without the pythagorean theorem May 21 accepted How come the Euclidian distance for n-space involves only squares? May 21 awarded Commentator May 21 comment How come the Euclidian distance for n-space involves only squares? Cool! Are there any uses to such alternative definitions? Also, what do you mean by saying that one allows rotation while the other does not? How can a distance metric prevent rotation? May 21 asked How come the Euclidian distance for n-space involves only squares? Mar 14 awarded Yearling Mar 6 revised What's so special about sine? (Concerning $y'' = -y$) corrected spelling, fixed formatting Mar 6 suggested approved edit on What's so special about sine? (Concerning $y'' = -y$) Mar 3 comment What's so special about sine? (Concerning $y'' = -y$) I understand that, but I don't completely buy this. Wherever there is a circle in nature, looking closer we find it not a circle. Orbits are actually ellipses, the earth isn't quite spherical, and so on. Mar 3 comment What's so special about sine? (Concerning $y'' = -y$) "Nearly every"? Which ones can't? Also, given the taylor series approximation for sine, can't anything approximated by a harmonic oscillator be modeled by even simpler functions? Feb 29 comment What's so special about sine? (Concerning $y'' = -y$) Thanks! I suppose I probably should take this question over to physics, I'm also curious as to why physics seems to care so much about the second derivative (in everything from F=ma to the position/momentum interaction). Feb 29 accepted What's so special about sine? (Concerning $y'' = -y$) Feb 29 awarded Nice Question Feb 28 awarded Editor Feb 28 revised What's so special about sine? (Concerning $y'' = -y$) added 448 characters in body Feb 28 asked What's so special about sine? (Concerning $y'' = -y$) Jun 6 comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? That helped a lot, thanks. In my own terms, the key point that I was missing was that taking $\lim_{n\to\infty}$ negates the drift off the unit circle that you see in $1+iX/n$, but does not negate the rotation caused by $1+iX/n$. It still astounds me that by taking the limit in this situation you negate the drift but not the rotation - that's an incredibly beautiful and powerful idea. :) Jun 6 accepted Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? Jun 6 comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? I get it! Multiplying by $1+ix$ gives us rotation by x but doesn't keep us on the unit circle. However, if we divide by n and raise to the power n, we manage to quell the drift off the circle but maintain the rotation. The drift is lost if we divide by and raise to the power n, but the rotation is not. Looking deeper, is there any field of math concerned with which operations are susceptible to elimination by such limits and which operations are unaffected?