Question

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Answer 1

$1)\; \; 2cos5t<\sqrt2\; \; \; \to \; \; cos5t<\frac{\sqrt2}{2}\\\\\frac{\pi}{4}+2\pi n<5t<\frac{7\pi }{4}+2\pi n \; ,\; \; \frac{\pi }{20}+\frac{2\pi n}{5}<t<\frac{7\pi}{20}+\frac{2\pi n}{5}\; ,\; n\in Z\\\\2)\; \; 2sin(-2t)<\sqrt3\; \; \to \; \; sin2t>-\frac{\sqrt3}{2}\\\\\frac{\pi }{3}+2\pi n<2t<\frac{2\pi}{3}+2\pi n\; \; ,\; \; \frac{\pi}{6}+\pi n<t<\frac{\pi}{3}+\pi n\; ,\; n\in Z\\\\3)\; \; cos3t>\frac{1}{3}\\\\-arccos\frac{1}{3}+2\pi n<3t<arccos\frac{1}{3}+2\pi n\; ,\; n\in Z\\\\-\frac{1}{3}arccos\frac{1}{3}+\frac{2\pi n}{3}<t<\frac{1}{3}arccos\frac{1}{3}+\frac{2\pi n}{3}\; ,\; n\in Z$

$4)\; \; 3sin(2t-\frac{\pi }{4})\leq 1\\\\ \sin(2t-\frac{\pi}{4})\leq \frac{1}{3}\\\\\pi -arcsin\frac{1}{3}+2\pi n<2t-\frac{\pi }{4}<2\pi +arcsin\frac{1}{3}+2\pi n\; ,\\\\\frac{5\pi }{4}-arcsin\frac{1}{3}+2\pi n<2t<\frac{9\pi }{4}+arcsin\frac{1}{3}+2\pi n\\\\\frac{5\pi }{8}-\frac{1}{2}arcsin\frac{1}{3}+\pi n<t<\frac{9\pi }{8}+\frac{1}{2}arcsin\frac{1}{3}+\pi n\; ,\; n\in Z$