Solve the following equations: + 2x+ 3=0 - Vidyadip

Question

Solve the following equations: $\sin\text{x}+\sin2\text{x}+\sin3=0$

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Answer

We have, $\sin\text{x}+\sin2\text{x}+\sin3=0$ $\Rightarrow\sin2\text{x}+2\sin2\text{x}.\cos\text{x}=0$ $\Rightarrow\sin2\text{x}+(1+2\cos\text{x})=0$ $\Rightarrow\text{Either}$ $\sin2\text{x}=0$ or $1+2\cos\text{x}=0$ $\Rightarrow2\text{x}=\text{n}\pi,\text{n}\in\text{z}$ or $\cos\text{x}=-\frac{1}{2}=\cos\Big(\pi-\frac{\pi}{3}\Big)$ $\Rightarrow\text{x}=\frac{\text{n}\pi}{2},\text{n}\in\text{z}$ or $\text{x}=2\text{m}\pi\pm\frac{2\pi}{3},\text{m}\in\text{z}$ Thus, $\text{x}=\frac{\text{n}\pi}{2},\text{n}\in\text{z}$ or $\text{x}=2\text{m}\pi\pm\frac{2\pi}{3},\text{m}\in\text{z} $

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