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Question Number 106468 by bemath last updated on 05/Aug/20

$$4\cos \mathrm{x}\cos 2\mathrm{x}\cos 3\mathrm{x}=1$$

Answered by john santu last updated on 05/Aug/20

$$\mathrm{recall}:2\cos 3\mathrm{x}\cos \mathrm{x}=\cos 4\mathrm{x}+\cos 2\mathrm{x}$$$$\Rightarrow 2\{\cos 4\mathrm{x}+\cos 2\mathrm{x}\}\cos 2\mathrm{x}=1$$$$2\{2\cos {}^{2}2\mathrm{x}-1+\cos 2\mathrm{x}\}\cos 2\mathrm{x}=1$$$$\mathrm{let}\cos 2\mathrm{x}=\vartheta$$$$\Rightarrow 2(2{\vartheta }^2+\vartheta -1)\vartheta -1=0$$$$4{\vartheta }^3+2{\vartheta }^2-2\vartheta -1=0$$$$(2\vartheta +1)(2{\vartheta }^2-1)=0$$$$\mathrm{case}\left(1\right)\cos 2\mathrm{x}=-\frac{1}{2}$$$$2\mathrm{x}=\pm \frac{2\pi }{3}+\mathrm{k}.360°;\mathrm{x}=\pm \frac{\pi }{3}+\mathrm{k}.180°$$$$\mathrm{case}\left(2\right)\cos 2\mathrm{x}=\frac{\sqrt{2}}{2}$$$$2\mathrm{x}=\pm \frac{\pi }{4}+\mathrm{k}.360°;\mathrm{x}=\pm \frac{\pi }{8}+\mathrm{k}.180°$$$$\mathrm{case}\left(3\right)\cos 2\mathrm{x}=-\frac{\sqrt{2}}{2}$$$$2\mathrm{x}=\pm \frac{3\pi }{4}+\mathrm{k}.360°;\mathrm{x}=\pm \frac{3\pi }{8}+\mathrm{k}.180°$$

Commented by bemath last updated on 05/Aug/20

$$\mathrm{thank}\mathrm{you}$$

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