Find a simple expression for the speed of light in a material in terms of \(c\) and the critical angle at an interface between the material and vacuum.

Snell's Law states that the ratio of the sine of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalently, to the reciprocal of the ratio of the indices of refraction. The formula is \(n_1 \sin(θ_1) = n_2 \sin(θ_2)\), where \(n_1, n_2\) are indices of refraction and \(θ_1, θ_2\) are the angles of incidence and refraction. The index of refraction of a material is defined as \(n = c/v\), where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the material. For the boundary between the material and vacuum, \(n_1 = n\), \(θ_1 = θ_c\) (the critical angle), \(n_2 = 1\) (since speed of light in vacuum is \(c\)) and \(θ_2 = 90^\circ\). Therefore, Snell's law at the critical angle becomes \(n \sin(θ_c) = 1\).

Substitute the relationship of \(n = c/v\) into the equation from step 1. This gives \(c/v \sin(θ_c) = 1\). Now, solve this equation for \(v\), the speed of light in the material. Dividing both sides by \(\sin(θ_c)\), you get \(c/v = \sin(θ_c)\). Finally, rearranging this equation gives \(v = c / \sin(θ_c)\).