specifically I have this equation
$n_p = \frac{n_A \sin(180 - \theta)}{\sin \left(60 - \sin^{-1}\left(\frac{n_A \sin(60)}{n_p}\right)\right)}$
how do I make $n_p$ the subject of the equation knowing that $0 < n_p < 2$ and $90 < \theta < 180$
wolfram alpha could solve it for me but wasnt able to produce the steps it took. I need to write an algorithm for my experiment.
I'm doing a physics experiment on prisms and snells law and ended up with this equation but lack the math ability to carry on with my experiment. Any help is appreciated.
The denominator is of the form $\sin(a+b) = \sin a \cos b + \cos a \sin b$
So the denominator can be expanded to:
$\sin 60 \cos(\arcsin x) - \cos 60 \sin \arcsin x$
where $x = \frac{n_A\sin 60}{n_p}$
Assuming all your angles are in degrees, $\sin 60 = \frac{\sqrt 3}{2}, \cos 60 = \frac 12$
Note that $\sin (\arcsin x) = x$ and $\cos (\arcsin x) = \sqrt{1-x^2}$.
So you can now rewrite the denominator as:
$\frac{\sqrt 3}{2} \sqrt{1-x^2} - \frac x2$
The numerator can be simplified a little by noting that $\sin(180 - \theta) = \sin \theta$, so the numerator becomes $n_A \sin \theta$
May I assume your basic algebraic skills are up to handling the rest?