Law of Cosines/Proof 3

Method 1

 * 1) Let $r^2=x^2+y^2$
 * $y^2=r^2+0-x^2$
 * Hence, $y^2=r^2-2xr(x/r)+x^2$


 * This result follows from the fact that $x=x\implies x-x=0\implies x^2-x^2=0$
 * And $x^2=xr(x/r)$

Method 2
Then $y^2=r^2+x^2-2xr\cos\theta$
 * 1) Let $r^2=x^2+y^2$
 * Double each side
 * Doubling each side is a consequence of repeated expansions of each variable, so doubling at the beginning is useful for reducing the proof's length
 * Then $2(r/x)=2(x/r)+2(y^2/xr)$
 * Then $2y^2=2r^2-2xr(x/r)$
 * Let $2r^2=r^2+x^2+y^2$ and $2y^2-y^2=r^2+x^2$

General Law of Cosines

 * 1) Let $C=2\pi r$
 * Let $r=(x^2+y^2)^\frac{1}{2}$ or $r^2=(\frac{C}{2\pi})^2$. For Illustrative purposes, the term including $\pi$ will be used.
 * Because $r^2=(\frac{C}{2\pi})^2$ and $r^2=x^2+y^2$ then $x^2+y^2=(\frac{C}{2\pi})^2$
 * Then $4\pi^2 x^2+4\pi^2 y^2=C^2$ but $C^2=(2\pi r)^2$. Let $4\pi^2 r^2=4\pi^2 x^2+4\pi^2 y^2$
 * Then execute Method 2 to derive the Law of Cosines for $y^2$