As θ takes all real values, how many integers lie in the range of 9sinθ + 40cosθ?

The expression sweeps the whole closed interval from −√(81 + 1600) = −41 to +41, since √1681 = 41. The integers available are −41, −40, …, −1, 0, 1, …, 41: that is 41 negative, 41 positive and zero, so 83 in all. Forgetting to count 0 gives 82, dropping both endpoints gives 81, and counting only the positive ones gives 41.