# Problem 2. (2 points)

The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take?

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# Answer: $2 ; 4$

## Solution:

Notice that $\sin x+\cos x=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)$. The argument of the sine function ranges from $\frac{\pi}{4}$ to $\frac{3 \pi}{4}$, so $\frac{\sqrt{2}}{2} \leqslant \sin \left(x+\frac{\pi}{4}\right) \leqslant 1$. Therefore, $1 \leqslant \sin x+\cos x \leqslant \sqrt{2}$, and the sum of five sines and five cosines