Well, we could replace cos(x) with sqrt(1 - sin^2(x)) and solve the resulting radical equation in sin(x); but that turns into a quartic equation that, with the irrational numbers involved, doesn't seem very workable.
So here's a new question: In your context, is it possible that you are expected to find approximate solutions using technology, rather than exact solutions by algebraic means? What else would I learn if I grabbed your book from you and looked through the context? Are any special methods needed for other problems nearby?