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Question Number 133321 by 777316 last updated on 21/Feb/21

$$Findx:$$$$sin\left(3x\right)-sin\left(2x\right)-2sin\left(x\right)=\sqrt{3}cos\left(x\right)$$

Commented by bramlexs22 last updated on 21/Feb/21

$$\mathrm{x}=\frac{2\pi }{3}$$$$\sin \left(3\times \frac{2\pi }{3}\right)-\sin \left(2\times \frac{2\pi }{3}\right)-2\sin \left(\frac{2\pi }{3}\right)=$$$$\sin \left(2\pi \right)-\sin \left(\frac{4\pi }{3}\right)-2\sin \left(\frac{2\pi }{3}\right)=$$$$0+\frac{\sqrt{3}}{2}-2\left(\frac{\sqrt{3}}{2}\right)=-\frac{\sqrt{3}}{2}\Leftarrow \mathrm{LHS}$$$$\mathrm{RHS}\Rightarrow \sqrt{3}\cos \left(\frac{2\pi }{3}\right)=\sqrt{3}(-\frac{1}{2})=-\frac{\sqrt{3}}{2}$$

Answered by bramlexs22 last updated on 21/Feb/21

$$\mathrm{x}=\frac{2\pi }{3},\frac{8\pi }{3},\frac{14\pi }{3},...$$

Commented by 777316 last updated on 21/Feb/21

how

Commented by bramlexs22 last updated on 21/Feb/21

Answered by MJS_new last updated on 21/Feb/21

$$\sin 3x-\sin 2x-\sin x=\sqrt{3}\cos x$$$$...$$$$\pm ({4\cos }^{2}x-2\cos x-3)\sqrt{1-{\cos }^{2}x}=\sqrt{3}\cos x$$$$\mathrm{let}c=\cos x$$$$\mathrm{squaring}\mathrm{and}\mathrm{transforming}$$$$c^6-c^5-\frac{9}{4}{c}^4+\frac{7}{4}{c}^3+2c^2-\frac{3}{4}c-\frac{9}{16}=0$$$$(c+\frac{1}{2})(c^5-\frac{3}{2}{c}^4-\frac{3}{2}{c}^3+\frac{5}{2}{c}^2+\frac{3}{4}c-\frac{9}{8})=0$$$$\mathrm{the}{5}^{\mathrm{th}}\mathrm{degree}\mathrm{polynome}\mathrm{has}\mathrm{one}\mathrm{real}\mathrm{solution}\mathrm{we}$$$$\mathrm{can}\mathrm{only}\mathrm{approximate}$$$$c_1=-\frac{1}{2}=\cos x_1$$$$c_2\approx .777106906075=\cos x_2$$$$\mathrm{testing}\mathrm{the}\mathrm{solutions}\left(\mathrm{because}\mathrm{we}\mathrm{have}\mathrm{squared}\right)$$$$\mathrm{leads}\mathrm{to}$$$$\Rightarrow$$$$x_1=\frac{2\pi }{3}+2n\pi$$$$x_2\approx -.680740465517+2n\pi$$

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