Thread: A nasty "solve for y(x)"

1. A nasty "solve for y(x)"

Given:

$\alpha+\beta=\frac{\pi}{2}+\frac{x}{2}$
$\sin(\alpha)=y$
$\tan^2(\beta)=\frac{a}{\lambda y^2}\frac{a-\lambda y^2}{a-1}$
$a=\frac{1}{2}\left(1+\lambda + \sqrt{(1+\lambda)^2-4\lambda y^2}\right)$,

solve for y(x) where $\lambda$ is a known parameter.

2. Originally Posted by Heirot
Given:

$\alpha+\beta=\frac{\pi}{2}+\frac{x}{2}$
$\sin(\alpha)=y$
$\tan^2(\beta)=\frac{a}{\lambda y^2}\frac{a-\lambda y^2}{a-1}$
$a=\frac{1}{2}\left(1+\lambda + \sqrt{(1+\lambda)^2-4\lambda y^2}\right)$,

solve for y(x) where $\lambda$ is a known parameter.

$\displaystyle \beta = \frac{\pi }{2} + \frac{x}{2} - alpha$
$cos(y) = \sqrt{1 - y^2}$