Find the general solution for each of the following equations: + … - Vidyadip

Question

Find the general solution for each of the following equations:
$\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$

✓

Answer

Given: $\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$
$\Rightarrow2\sin\Big(\frac{5\text{x}+\text{x}}{2}\Big)\cos\Big(\frac{5\text{x}-\text{x}}{2}\Big)+\sin3\text{x}=0$
$\Rightarrow2\sin3\text{x}\cos2\text{x}+\sin3\text{x}=0$
$\Rightarrow\sin3\text{x}(2\cos2\text{x}+1)=0$
$\Rightarrow\sin3\text{x}=0$ or $2\cos2\text{x}+1=0$
$\Rightarrow3\text{x}=\text{n}\pi$ or $\cos2\text{x}=\frac{-1}{2}=\cos\frac{2\pi}{3},\text{n}\in\text{Z}$
$\Rightarrow\text{x}=\frac{\text{n}\pi}{3}$ where $2\text{x}=2\text{n}\pi\pm\frac{2\pi}{3},\text{n}\in\text{Z}$
$\Rightarrow\text{x}=\frac{\text{n}\pi}{3}$ where $\text{x}=\text{n}\pi\pm\frac{\pi}{3},\text{n}\in\text{Z}$

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