Solve the following equations: + 5 = 3 - Vidyadip

Question

Solve the following equations: $\sin\theta+\sin5\theta=\sin3\theta$

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Answer

$\sin\text{x}+\sin5\text{x}=\sin3\text{x}$ $\Rightarrow2\sin3\text{x},\cos2\text{x}-\sin3\text{x}=0$ $\Big[\because\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C}+\text{D}}{2}.\cos\frac{\text{C}-\text{D}}{2}\Big] $ Either $\Rightarrow\sin3\text{x}=0$ or $2\cos2\text{x}-1=0 $ $\Rightarrow3\text{x}=\text{n}\pi,\text{n }\in\ \text{z}$ or $\cos2\text{x}=\frac{1}{2}=\cos\frac{\pi}{3}$ $\Rightarrow\text{x}=\frac{\text{n}\pi}{3},\text{n }\in\ \text{z}$ or $2\text{x}=2\text{m}\pi\pm\frac{\pi}{3},\text{m }\in\ \text{z}$ or $\text{x}=\text{m}\pi\pm\frac{\pi}{6}$ Thus, $\text{x}=\frac{\text{n}\pi}{3}$ or $\text{m}\pi\pm\frac{\pi}{6},\text{n, m }\in\ \text{z}$

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