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)

= Unit circle def'n = Just getting the skeleton of the definition out, I'll work on it later.

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

$\sin \theta := y$

$\cos \theta := x$

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



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)

=Carroll Paradox=

To be proven: $q$

1. Assume $p \implies q$.

2. Assume $p$.

3. $p \land (p \implies q) \vdash q$.

4. From 1 and 2, $p \land (p \implies q)$.

5. $(p \land (p \implies q) \land (p \land (p \implies q) \vdash q)) \vdash q$.