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Question Number 107304 by bemath last updated on 10/Aug/20

$$⋎bemath⋎$$$$\left(1\right)Finddomainoffunction$$$$f\left(x\right)=\sqrt{\log {}_{0.2}\left(\frac{x+2}{x-1}\right)-1}$$$$\left(2\right)\frac{\sin {}^2(\frac{9\pi }{8}-2x)-\sin {}^2(\frac{7\pi }{8}-2x)}{\sin (2018\pi +4x)}=?$$

Answered by bobhans last updated on 10/Aug/20

$$✠\mathbf{bobhans}✠$$$$\left(1\right){\mathrm{D}}_{\mathrm{f}}:\log {}_{0.2}\left(\frac{\mathrm{x}+2}{\mathrm{x}-1}\right)-1⩾0$$$$\log {}_{0.2}\left(\frac{\mathrm{x}+2}{\mathrm{x}-1}\right)⩾\log {}_{0.2}\left(\frac{1}{5}\right)$$$$\frac{\mathrm{x}+2}{\mathrm{x}-1}⩽\frac{1}{5}\Rightarrow \frac{5\mathrm{x}+10-\mathrm{x}+1}{5(\mathrm{x}-1)}⩽0$$$$\frac{4\mathrm{x}+11}{5(\mathrm{x}-1)}⩽0\Rightarrow -\frac{11}{4}⩽\mathrm{x}<1...\left(1\right)$$$$\Rightarrow \frac{\mathrm{x}+2}{\mathrm{x}-1}>0\Rightarrow \mathrm{x}<-2\cup \mathrm{x}>1$$$$\mathrm{solution}\Rightarrow \left(1\right)\cap \left(2\right):\mathrm{x}⩽-\frac{11}{4}$$

Answered by bobhans last updated on 10/Aug/20

$$⌢\mathcal{B}\mathrm{obhans}⌢$$$$\left(2\right)\frac{(\sin (\frac{9\pi }{8}-2\mathrm{x})+\sin (\frac{7\pi }{8}-2\mathrm{x}))(\sin (\frac{9\pi }{8}-2\mathrm{x})-\sin (\frac{7\pi }{8}-2\mathrm{x}))}{\sin 4\mathrm{x}}=$$$$\frac{\left(2\sin (\pi -2\mathrm{x})\cos \left(\frac{\pi }{8}\right)\right)\left(2\cos (\pi -2\mathrm{x})\sin \left(\frac{\pi }{8}\right)\right)}{\sin 4\mathrm{x}}=$$$$\frac{\sin \left(\frac{\pi }{4}\right)\left(\sin (2\pi -4\mathrm{x})\right)}{\sin 4\mathrm{x}}=\frac{\frac{\sqrt{2}}{2}.\sin 4\mathrm{x}}{\sin 4\mathrm{x}}=\frac{\sqrt{2}}{2}$$

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