2002 AIME II Problem 10

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of x for which the sine of x degrees is the same as the sine of x radians are \frac{m \pi}{n-\pi} and \frac{p \pi}{q+\pi}, where m, n, p and q are positive integers. Find m+n+p+q.