The thing is, that sin 72 = sin 108 is [like] a *principium individuationis *[spelling?] of the general situation. If you look at the horizontal and vertical axes, the sine of an angle in the first quadrant (which 72 degrees is) is reflected in the vertical axes in the second quadrant at sin 108. 180 degrees is the horizontal axis. Subtract 72 from from 180, you have your 72.

In general, sin(180 - theta) = sin(theta)

and cos(180 - theta) = - cos(theta)

You can see why. Sine is positive in both first and second quadrants, whereas cosine is negative in second quadrant (since it is to the left of origin).

Take care, brother seeker. You wish to better understand, and to see the general in the particular. That is a good motivation that makes your quest more authentic.