When an electromagnetic wave strikes an interface between two linear media, Snell's law states that $\frac{\sin(\theta_T)}{\cos(\theta_I)} = \frac{n_1}{n_2}$ where $\theta_I$ is the angle of incidence, $\theta_T$ is the angle of transmission, $n_1$ is the index of refraction of the first medium, and $n_2$ is the index of refraction of the second medium.
In the case where $n_2 > n_1$, we can then see that $\sin(\theta_T) = \frac{n_1}{n_2} \sin(\theta_I)$
From this we can derive a critical angle $\theta_C$ such that when $\theta_I = \theta_C$ we have $\theta_T = 90 \deg$. Consider, for instance, the case where light travels from water with an index of refraction of $n_1=1.35$ and air with an index of refraction of $n=1$. Then we find that $\theta_C=47.8 \deg$.
By taking $\theta_I = \theta_C + \varepsilon$ where $\varepsilon$ is some small positive value, we find $\sin(\theta_T) = \frac{1.35}{\theta_C+\epsilon} > 1$. That is, the sin function is taking a value outside of it's range!
Normally I would chalk this up to the problem being "out of the bounds" of the mathematical model of Snell's law, but Griffiths uses this fact to derive evanescent waves:
The only change is that $\sin(\theta_T) = \frac{n_1}{n_2} \sin(\theta_I)$ is now greater than $1$, and $\cos(\theta_T) = \sqrt{1-\sin^2(\theta_T)} = i\sqrt{\sin^2(\theta_T)-1}$ is imaginary. (Obviously, $\theta_T$ can no longer be interpreted as an angle!)
How is it possible that $\cos(\theta_T)$ is imaginary? What does it mean that $\theta_T$ cannot be interpreted as an angle?
