Only a real mathematician (or future mathmetician) can appreciate finding exact values like this - regardless of how ugly they may look.
True enough; but what we then want to do is to generalize in order to understand the deeper patterns. So from Wikipedia:
"According to Baker's theorem, if the value of a sine, a cosine or a tangent is algebraic, then either the angle is rational number of degrees, or the angle is a transcendental number of degrees. That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values."
I don't know if there is a quick way of obtaining these answers for all of the sin(7kπ) values but I just formed the equation above, and solved it using the cubic formula. If there is a quicker way of doing these (maybe a general formula or something similar) please let me know.