Question

$\sqrt{2}$(sin2x + sinx) -2cosx $\leq$ 1 , если x ∈ [0; π]

Answer 1

$\sqrt2(sin2x+sinx)-2cosx\leq 1\ \ ,\ \ \ x\in [\ 0\ ;\ \pi \ ]\\\\\sqrt2\cdot 2sinx\cdot cosx+\sqrt2sinx-2cosx-1\leq 0\\\\\sqrt2sinx\cdot (2cosx+1)-(2cosx+1)\leq 0\\\\(2cosx+1)(\sqrt2sinx-1)\leq 0\\\\a)\ \ \left\{\begin{array}{l}2cosx+1\leq 0\\\sqrt2sinx-1\geq 0\end{array}\right\ \ \ ili\ \ \ b)\ \ \left\{\begin{array}{l}2cosx+1\geq 0\\\sqrt2sinx-1\leq 0\end{array}\right$

$a)\ \left\{\begin{array}{l}cosx\leq -\dfrac{1}{2}\\sinx\geq \dfrac{1}{\sqrt2}\end{array}\right\ \ \ \ \left\{\begin{array}{l}\dfrac{2\pi}{3}+2\pi n\leq x\leq \dfrac{4\pi }{3}+2\pi n\ ,\ n\in Z\\\dfrac{\pi}{4}+2\pi k\leq x\leq \dfrac{3\pi }{4}+2\pi k\ ,\ k\in Z\end{array}\right\\\\\\\dfrac{2\pi }{3}+2\pi n\leq x\leq \dfrac{3\pi}{4}+2\pi n\ ,\ \ n\in Z$

$b)\ \left\{\begin{array}{l}cosx\geq -\dfrac{1}{2}\\sinx\leq \dfrac{1}{\sqrt2}\end{array}\right\ \ \ \ \left\{\begin{array}{l}-\dfrac{2\pi}{3}+2\pi n\leq x\leq \dfrac{2\pi }{3}+2\pi n\ ,\ n\in Z\\-\dfrac{5\pi}{4}+2\pi k\leq x\leq \dfrac{\pi }{4}+2\pi k\ ,\ k\in Z\end{array}\right\\\\\\-\dfrac{2\pi }{3} \leq x\leq \dfrac{\pi}{4}+2\pi n\ ,\ n\in Z$

$c)\ \ x\in \Big[-\dfrac{2\pi }{3}+2\pi n\ ;\ \dfrac{\pi}{4}+2\pi n\ \Big]\cup \Big[\ \dfrac{2\pi }{3}+2\pi n\ ;\ \dfrac{3\pi }{4}+2\pi n\ \Big]\ \ ,\ n\in Z\ -\ otvet\\\\\\d)\ \ x\in [\ 0\ ;\ \pi \ ]:\ \ x\in \Big[\ 0\ ;\, \dfrac{\pi}{4}\ \Big]\cup \Big[\ \dfrac{2\pi }{3}\ ;\ \dfrac{3\pi }{4}\ \Big]$