TITLE: Sine and cosine expression
QUESTION [7 upvotes]: Find the greatest possible value of $5\cos x + 6\sin x$.
I attempted to solve this using graphing, however, the answer appears to be an ugly irrational. Is there a better method of solving this problem?
Thank you.

REPLY [14 votes]: A simple vector-based approach: you recognize in the expression the dot product $$(5, 6) \cdot (\cos x, \sin x).$$ This product is maximized when the two vectors are parallel and it is then the product of the moduli $$\sqrt{5^2+6^2} \cdot 1.$$