Find the general solution for each of the following equations: 2x… - Vidyadip

Question

Find the general solution for each of the following equations:
$\sin2\text{x}+\cos\text{x}=0$

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Answer

$\sin2\text{x}+\cos\text{x}=0$$\Rightarrow\sin2\text{x}+\cos\text{x}+\cos\text{x}=0$
$\Rightarrow\cos\text{x}(2\sin\text{x}+1)=0$
$\Rightarrow\cos\text{x}=0$ or $2\sin\text{x}+1=0$
Now, $\cos\text{x}=0\Rightarrow\cos\text{x}=(2\text{n}+1)\frac{\pi}{2},$ where $\text{n}\in\text{Z}$
$2\sin\text{x}+1=0$
$\Rightarrow\sin\text{x}=\frac{-1}{2}=-\sin\frac{\pi}{6}=\sin\Big(\pi+\frac{\pi}{6}\Big)$
$=\sin\Big(\pi+\frac{\pi}{6}\Big)=\sin\frac{7\pi}{6}$
$\Rightarrow\text{x}=\text{n}\pi+(-1)^\text{n}\frac{7\pi}{6},$ where $\text{n}\in\text{Z}$
Therefore, the general solution is $(2\text{n}+1)\frac{\pi}{2}$ or $\text{n}\pi+(-1)^2\frac{7\pi}{6},\text{n}\in\text{Z}$

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