Find the general solutions of the following equations: 9x= - Vidyadip

Question

Find the general solutions of the following equations: $\sin9\text{x}=\sin\text{x}$

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Answer

$\sin9\text{x}=\sin\text{x}$ $\sin9\text{x}-\sin\text{x}=0$ Apply $\sin\text{A}-\sin\text{B}$ Formula $\sin\text{A}-\sin\text{B}=2\cos\Big(\frac{\text{A+B}}{2}\Big)\sin\Big(\frac{\text{A-B}}{2}\Big)$ $\sin9\text{x}-\sin\text{x}2\cos5\text{x}\sin4\text{x}=0$ $\cos5\text{x}\sin4\text{x}=0$ $\Rightarrow\cos5\text{x}=0$ or $\sin4\text{x}=0$ $5\text{x}=\frac{(2\text{n}+1)\pi}{2}$ (or) $4\text{x}=\text{n}\pi$ $\text{x}=\Big\{\frac{(2\text{n}+1)\pi}{10}\Big\}$ {or}$\text{x}=\Big\{\frac{\text{n}\pi}{4}\Big\}$where$\text{n}\in\text{z}$

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