Question: hi i am working on a physics problem but got...

Hi, I am working on a physics problem but got stuck on the algebra part of the solution, can someone please help me with simultaneous equations?

I am applying Snell's Law to find the refractive index of a prism ($n_2$), however, the only information I have is:

${\theta }_1=37°$

${\theta }_4=90°$

$n_1=1$

And the two equations would be:

$Eq. 1 - n_1\sin \left({\theta }_1\right)=n_2\sin \left({\theta }_2\right) => 1*\sin \left(37\right)=n_2\sin \left({\theta }_2\right) => \sin \left(37\right)=n_2\sin \left(60-{\theta }_3\right) => n_2=\frac{\sin \left(37\right)}{\sin \left(60-{\theta }_3\right)}$$Eq. 2 - n_2\sin \left({\theta }_3\right)=n_1\sin \left({\theta }_4\right) => n_2\sin \left({\theta }_3\right)=1*\sin \left(90\right) => n_2\sin \left({\theta }_3\right)=1 => n_2=\frac{1}{\sin \left({\theta }_3\right)}$

This was an optics lab session and I am using a method called total internal reflection and then finding the refraction index from the critical angle (which is ${\theta }_3$ in this problem). After that point I used trig identities to resolve $\sin \left(60-{\theta }_3\right)$ but then I got stuck and not sure if I equated the equations properly.

Thank you very much,

Michele