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 Jun 10 awarded Yearling Jun 10 comment Why do we write second derivatives like $\frac{d^2x}{dt^2}$ This makes very little sense to me. How come we can separate the $d$ from $x$ in $dx$, but we can't separate the $d$ from $t$ in $dt$? What are these $d$ things anyway, and why do we treat the one attached to $x$ differently from the one attached to $t$? Jun 10 asked Why do we write second derivatives like $\frac{d^2x}{dt^2}$ May 23 awarded Popular Question Oct 28 awarded Popular Question May 23 revised Generalizing an alternative derivation of distance added 85 characters in body May 22 comment Generalizing an alternative derivation of distance @JonasMeyer you're right, it's independent of the cosine rule and just dependent upon the definition of cosine -- I just didn't notice it until I was working through the derivation of the Pythagorean theorem from the cosine rule alone. May 22 revised Generalizing an alternative derivation of distance added 9 characters in body May 22 comment Generalizing an alternative derivation of distance @MarkDominus No problem. The reason I'm interested is a more-than-600-character rabbit hole. Succinctly, I'm wondering why the Pythagorean theorem occurs. (Why are squares so closely related to measuring distance? What do the squares of the sides have to do with anything?) When I ask, people tend to show me proofs, which is the "how", not the "why", so I'm working through it from another direction. To me it seems that the Pythagorean theorem is emergent from a special case of the cosine rule and that projecting sides onto the axis of distance is a more fundamental way of measuring distance. May 22 comment Generalizing an alternative derivation of distance @QiaochuYuan: d=(√x2+y2) basically says "in order to find d, extrapolate x and y into squares, mush the squares together into one big square, and then measure a side of the new square." I understand why it works, but it seems a coincidental (not fundamental) way to find distance. (Whereas xcos(Y)+ycos(X) is the projection of the x and y axes onto the axis of the distance, that's a very straightforward way of finding it.) Regardless, I'm not asking you to empathize with my curiosity: merely to help me out with my questions. May 22 comment Generalizing an alternative derivation of distance @MarkDominus I'm not asking you to understand why I prefer my approach. May 22 comment Generalizing an alternative derivation of distance @QiaochuYuan it seems unnatural to me that triangular distances are found by extrapolating squares, adding them, and then rooting them. I'm trying to do the same thing but with trig. So by "equivalent" I mean "identical given Cartesian coordinates." By "generalized" I mean this: The Pythagorean theorem works in three-dimensional space. How do you add a z dimension to $x*cos(tan^{-1}(y/x)) + y*cos(tan^{-1}(x/y))$? May 22 comment Generalizing an alternative derivation of distance @MarkDominus I disagree. It depends which one you take as fundamental. Obviously trigonometry and squares are closely related, and obviously you can always transform one into the other, but I'd like to try to derive the Pythagorean theorem from the trig instead of the (more common) inverse. But yes, obviously, the above should reduce to a generalized case of the Pythagorean theorem. May 22 asked Generalizing an alternative derivation of distance May 22 comment Can you show how triangles are related to harmonic growth without using sine? I'm just wondering what happens when you generalize to the case where instead of holding the length of the pole (and the 90* angle with the ground) constant, you hold the length of the opposite angle to the ground constant. It seems that, with this method, all triangles (varying one angle and holding another constant) trace out an ellipse, but I'm not sure how to go about proving or disproving such an idea. May 22 accepted Can you show how triangles are related to harmonic growth without using sine? May 22 comment Can you show how triangles are related to harmonic growth without using sine? Very slick, I like it. However, the x^2+y^2 = constant part is dependent upon the Pythagorean theorem, and I don't know of any proof that doesn't (at least implicitly) use sine. Is there one? Is there a way to prove that the motion is harmonic even when it's not a right triangle (where, in the example above, you'd explicitly have to use the cosine rule)? May 21 asked Can you show how triangles are related to harmonic growth without using sine? May 21 accepted How to derive the law of cosines without the pythagorean theorem May 21 comment How to derive the law of cosines without the pythagorean theorem Beautiful. Could you post it here so I can credit the answer?