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Question Number 144828 by mathlove last updated on 29/Jun/21

$$2\sin 17x+\sqrt{3}\cos 5x+\sin 5x=0$$$$x=?$$

Answered by liberty last updated on 30/Jun/21

$$2\sin 17\mathrm{x}+2\cos (5\mathrm{x}-\frac{\pi }{6})=0$$$$\Rightarrow \cos (5\mathrm{x}-\frac{\pi }{6})=-\sin 17\mathrm{x}$$$$\Rightarrow \cos (5\mathrm{x}-\frac{\pi }{6})=\cos (\frac{\pi }{2}+17\mathrm{x})$$$$\Rightarrow 2\mathrm{k}\pi \pm (5\mathrm{x}-\frac{\pi }{6})=\frac{\pi }{2}+17\mathrm{x}$$$$\Rightarrow 2\mathrm{k}\pi \mp \frac{\pi }{6}-\frac{\pi }{2}=17\mathrm{x}\mp 5\mathrm{x}$$$$\Rightarrow 2\mathrm{k}\pi -\frac{2\pi }{3}=17\mathrm{x}\mp 5\mathrm{x}$$$$\left(\mathrm{case}1\right)2\mathrm{k}\pi -\frac{2\pi }{3}=12\mathrm{x}$$$$\Rightarrow \mathrm{x}=\frac{\mathrm{k}\pi }{6}-\frac{\pi }{18}$$$$\left(\mathrm{case}2\right)2\mathrm{k}\pi -\frac{2\pi }{3}=22\mathrm{x}$$$$\Rightarrow \mathrm{x}=\frac{\mathrm{k}\pi }{11}-\frac{\pi }{33}.$$$$\left(\mathrm{case}3\right)2\mathrm{k}\pi -\frac{\pi }{3}=12\mathrm{x}$$$$\Rightarrow \mathrm{x}=\frac{\mathrm{k}\pi }{6}-\frac{\pi }{36}$$$$\left(\mathrm{case}4\right)2\mathrm{k}\pi -\frac{\pi }{3}=22\mathrm{x}$$$$\Rightarrow \mathrm{x}=\frac{\mathrm{k}\pi }{11}-\frac{\pi }{66}$$

Commented by liberty last updated on 30/Jun/21

$$\mathrm{o}\mathrm{yes}.\mathrm{thanks}$$

Answered by ajfour last updated on 29/Jun/21

$$\sin 17x+\sin (5x+\frac{\pi }{3})=0$$$$\Rightarrow 5x+\frac{\pi }{3}=n\pi -{(-1)}^n\left(17\right)x$$$$\Rightarrow x=\frac{n\pi -\frac{\pi }{3}}{5+17{(-1)}^n}$$

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