User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Thoughts on the picture, anyone? --GFauxPas 14:37, 9 November 2011 (CST)


 * Looks okay to me. I was going to get round to doing something similar myself at one point.


 * Mind, if you're planning on using it to illustrate sine and cosine, you might want to add the actual distances as lines. Oh, and purists may wince when they see x and y used to define the axes and the point on it, but I wouldn't be too fussed. --prime mover 14:47, 9 November 2011 (CST)

Try 2. I see I lost too many colors by saving it as a .gif, try 3 will be a .png or something.

File:Unitcirclev2.gif

Let $P = (x,y)$ be a point on the unit circle centered at the origin.

Let $\theta$ be the angle formed by the arc $(1,0)$, $(x,y)$ subtending the origin, measured counterclockwise.

The unit circle definition of the trigonometric functions are

$\cos \theta := x$

$\sin \theta := y$

That is, the directed distance between $P$ and the $x$-axis is the cosine, and the directed distance between $P$ and the $y$-axis is the sine.

Sources: khan academy "tau versus pi", wolfram mathworld "trigonometry"

After this is set up I can do a proof of the consistency between the right triangle definition and the circle definition. --GFauxPas 07:22, 17 November 2011 (CST)

Points On a Line
The points on an infinite straight line are in one-to-one correspondence with $\R$.

Okay the talk page for Real Number Line has all these complicated things that I don't understand. But I'm reading Tarski's book and he has a proof, and it makes me think that he's being very informal if you math guys can't do it straightforwardly. His proof is more or less like this, though all of the proofs aren't as rigorous as MW's, they're more conceptual, and he uses different terminology.

Part 1: Find a mapping from the line to $\R$.

Pick a starting point. Map it to zero.

Pick a different point. The directed distance between them is a non-zero real number.

Part 2: Find a mapping from $\R$ to the line.

Map 0 to a point on the line, that's the origin. Pick a number. If it's positive, draw a line with that length to the right of the origin. If it's negative, draw a line with that length to the left of the origin.

??? --GFauxPas 17:25, 19 November 2011 (CST)