Discussion
For \(0 \le x \le 2\pi\), \(\sin x\) and \(\cos x\) are both decreasing functions in the interval ________.
Answer: B
Both functions decrease on \((\frac{\pi}{2}, \pi)\).
On \([0, 2\pi]\), \(\sin x\) increases up to \(\pi/2\), decreases from \(\pi/2\) to \(3\pi/2\), then increases again. \(\cos x\) decreases from 0 to \(\pi\) and increases from \(\pi\) to \(2\pi\). The derivatives say the same: \((\sin x)' = \cos x\) is negative on \((\pi/2, 3\pi/2)\), and \((\cos x)' = -\sin x\) is negative on \((0, \pi)\).
The two decreasing intervals overlap only on \((\pi/2, \pi)\). On \((0, \pi/2)\) sine is increasing; on \((\pi, 3\pi/2)\) cosine is increasing; on \((3\pi/2, 2\pi)\) both are increasing.
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