This regards the following problem:
A ray of light is traveling in glass and strikes a glass/liquid interface. The angle of incidence is $58.0^\circ$ and the index of refraction of the glass is $n = 1.50$. What is the largest index of refraction that the liquid can have, such that none of the light is transmitted into the liquid and all of it is reflected back into the glass?
My solution: We have $n_1 \sin \theta_1 = n_2 \sin \theta_2$. We wish to maximize $n_2$ provided $90^\circ \leq \theta \leq 180^\circ$. Plugging in our givens, we have $(1.50)(\sin 58^\circ) = n_2 \sin \theta_2 \rightarrow n_2 = \frac{(1.50)(\sin 58^\circ)}{\sin \theta_2}$. Since we wsh to maximize $n_2$, we should minimize $\sin \theta_2$, which can be any value in $(0, 1]$ so there is no theoretical limit on the size of $n_2$ (according to this math)
However, the textbook says the correct maximum is $1.27$, which is taking $\theta_2 = 90^\circ$. I don't think this is right at all, since if anything that is the minimum. Consider $1 < n_2 < 1.27$. Then $1.27 > \sin \theta_2 > 1$, which is impossible given the range of the sine function over the reals.
Who is right?