Answer
The index of refraction of the glass is $~~1.60$
Work Step by Step
We can find an expression for $tan~\theta_B$:
$\theta_B = tan^{-1}~\frac{n_2}{n_1}$
$\theta_B = tan^{-1}~\frac{n_2}{1.00}$
$\theta_B = tan^{-1}~n_2$
$tan~\theta_b = n_2$
Note that the incident angle $\theta_1 = \theta_B$
We can use Snell's law to find an expression for $sin~\theta_B$:
$n_1~sin~\theta_1 = n_2~sin~\theta_2$
$1.00~sin~\theta_1 = n_2~sin~\theta_2$
$sin~\theta_B = n_2~sin~32.0^{\circ}$
$sin~\theta_B = 0.53~n_2$
By trigonometry:
$0.53~n_2 = \frac{n_2}{\sqrt{n_2^2+1}}$
$\sqrt{n_2^2+1} = \frac{n_2}{0.53~n_2}$
$\sqrt{n_2^2+1} = \frac{1}{0.53}$
$n_2^2+1 = (\frac{1}{0.53})^2$
$n_2^2 = (\frac{1}{0.53})^2-1$
$n_2 = \sqrt{(\frac{1}{0.53})^2-1}$
$n_2 = 1.60$
The index of refraction of the glass is $~~1.60$
