Question

Sin^2(x)>1/4
Cos^2(x)≤1/4
Розв'яжіть обидва нерівності окремо

Answer 1

$1)\; \; sin^2x>\dfrac{1}{4}\\\\\\\dfrac{1-cos2x}{2}>\dfrac{1}{4}\; \; ,\; \; 1-cos2x>\dfrac{1}{2}\; \; ,\; \; cos2x<\dfrac{1}{2}\; ,\\\\\\\dfrac{\pi}{3}+2\pi n<2x<\dfrac{5\pi}{3}+2\pi n\; ,\; n\in Z\\\\\\\dfrac{\pi}{6}+\pi n<x<\dfrac{5\pi}{6}+\pi n\; ,\; n\in Z\\\\\\x\in \Big(\dfrac{\pi}{6}+\pi n\; ;\; \dfrac{5\pi}{6}+\pi n\Big)\; ,\; n\in Z$

$2)\; \; cos^2x\leq \dfrac{1}{4}\\\\\\\dfrac{1+cos2x}{2}\leq \dfrac{1}{4}\; \; ,\; \; 1+cos2x\leq \dfrac{1}{2}\; \; ,\; \; cos2x\leq -\dfrac{1}{2}\; ,\\\\\\\dfrac{2\pi}{3}+2\pi n\leq 2x\leq \dfrac{4\pi}{3}+2\pi n\; ,\; n\in Z\\\\\\\dfrac{\pi}{3}+\pi n\leq x\leq \dfrac{2\pi}{3}+\pi n\; ,\; n\in Z\\\\\\x\in \Big[\; \dfrac{\pi}{3}+\pi n\; ;\; \dfrac{2\pi}{3}+\pi n\; \Big]\; ,\; n\in Z$