Perhaps so, but in this case the solution is one that's already well known (that you can express trig functions exactly when you're dealing with right triangles

Perhaps so, but in this case the solution is one that's already well known (that you can express trig functions exactly when you're dealing with right triangles with integer sides). If you want to do some computation involving a given angle, you're not likely to have the luxury of picking the angle that makes the computation easy using rational arithmetic. The sine of 45° is an irrational number (1/sqrt(2)).