While I was doing some problems on Geometrical Optics (Physics), in one of them, I was asked to find the dependence of angle of deviation $\delta$ with the angle of incidence $i$ for $0^\circ<I<90^\circ$, when a beam of light is incident from glass (refractive index $\mu=1.5$) to air.
For angle of incidence less than the critical angle, using Snell's law I determined the relation to be:
$$\delta = \sin^{-1}(\mu\sin i)-i$$
And for angle of incidence larger than the critical angle, using laws of reflection, I determined the relation to be:
$$\delta=180^\circ-2i$$
I tried to plot the variation of $\delta$ with $i$. The second function was very easy to plot. But without the use of a graphing calculator, I was unable to plot the first function $y=\sin^{-1}(k\sin x)-x$. Using a graphing calculator I plotted the above two functions as shown below:
Click here to open the graph in Desmos
It can be seen that the first function is initially linear then becomes concave upwards. Is it possible to plot the first one with a good amount of accuracy without using a graphing calculator?
Further, it can be seen that the graph is not continuous at the critical angle. Initially, I expected to get a continuous curve but it's discontinuous. I can understand from the two functions that discontinuity exists. But a particular value of angle of deviation exists when the ray is incident at critical angle, and we could not ascertain its value from the graph. Or in other words, is it possible to find the angle of deviation when $i$ is equal to the critical angle using only the graph?
I think this question is more of Mathematics and less of Physics and that's why I asked here. If you think this question belongs to Physics.SE, kindly intimate me in the comments, and I'll delete this one. Thank you.