Question

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Answer 1

$1)\ \ sin(3x-\dfrac{\pi}{4})=\dfrac{\sqrt3}{2}\\\\3x-\dfrac{\pi}{4}=(-1)^{n}\cdot \dfrac{\pi}{3}+\pi n\ \ ,\ \ 3x=(-1)^{n}\cdot \dfrac{\pi}{3}+\dfrac{\pi}{4}+\pi n\ ,\\\\x=(-1)^{n}\cdot \dfrac{\pi}{9}+\dfrac{\pi}{12}+\dfrac{\pi n}{3}\ ,\ n\in Z$

$2)\ \ cos3x=-\dfrac{5}{3}\ \ ,\ \ \ \ \ \ \ -\dfrac{5}{3}<-1\ \ ,\ \ -1\leq cos3x\leq 1\ \ \Rightarrow \ \ x\in \varnothing$

$3)\ \ 2cos^2x+cosx-1=0\\\\t=cosx\ ,\ \ -1\leq t\leq 1\ \ \ \to \ \ \ 2t^2+t-1=0\ \ ,\ \ D=9\ ,\ t_1=-1\ ,\ t_2=\dfrac{1}{2}\\\\cosx=-1\ \ \to \ \ \ x=\pi +2\pi n\ ,\ n\in Z\\\\cosx=\dfrac{1}{2}\ \ \to \ \ \ x=\pm \dfrac{\pi}{3}+2\pi k\ ,\ k\in Z$

$4)\ \ sinx\geq -\dfrac{\sqrt2}{2}\\\\\\-\dfrac{\pi }{4}+2\pi n\leq x\leq \dfrac{5\pi}{4}+2\pi n\ ,\ n\in Z\\\\\\x\in \Big(-\dfrac{\pi }{4}+2\pi n\ ;\ \dfrac{5\pi}{4}+2\pi n\ \Big)\ ,\ n\in Z$

$5)\ \ cosx\leq \dfrac{\sqrt3}{2}\\\\\\\dfrac{\pi }{6}+2\pi n\leq x\leq \dfrac{11\pi}{6}+2\pi n\ ,\ n\in Z\\\\\\x\in \Big(\dfrac{\pi }{6}+2\pi n\ ;\ \dfrac{11\pi}{4}+2\pi n\ \Big)\ ,\ n\in Z$