Most would have memorized the exact trigonometric values for those special angles (30o,45o,60o), but these angles are only special due to the convenience of the geometry of the isosceles triangle. If I were asked to find the value of, say, cos30o without use of these triangles, I would be out of luck doing so. With some study in complex numbers, our class was able to find the trig values for 15o and 75o by a method other than sin(A+B) ~ cos(A+B). So now we know all the trig values for angles that are multiples of 15o, which I see as being a large restriction. Eventually, by some unconventional way using complex polynomials, we found the trig values to 50o. This was a great breakthrough for us, but it is seemingly impossible to try and apply the same method to find other obscure angles. Is it possible to endeavour in finding the trig values (mostly sine and cosine) for any angles?