Problem of the Fortnight #6

Mr. Spock’s Favorite Angles

As we all know very well, Mr. Spock is/was/will be a very logical individual who prized rationality above all else. As Science Officer aboard the Enterprise, Mr. Spock was also a great fan of mathematics – and who could possibly blame him?! Thus, it is no surprise that Mr. Spock developed a deep and rational love for those angles between 0 and 90 degrees (or 0 and pi/2 radians) whose sine and cosine are both rational. Show that there are infinitely many such angles.

Solutions are due by 4pm on Friday, November 19^th in Goodrich 108