Find the general solutions of the following equations: 2x= 3x - Vidyadip

Question

Find the general solutions of the following equations:
$\sin2\text{x}=\cos3\text{x}$

✓

Answer

We have,
$\sin2\text{x}=\cos3\text{x}$
$\Rightarrow\cos3\text{x}=\cos\Big(\frac{\pi}{2}-2\text{x}\Big)\Big[\because\cos\Big(\frac{\pi}{2}-\text{x}\Big)=\sin\text{x}\Big]$
$\Rightarrow3\text{x}=2\text{n}\pi\pm\Big(\frac{\pi}{2}-2\text{x}\Big)\text{n}\in\text{z}$
$\Rightarrow\text{ Either}$
$5\text{x}=2\text{n}\pi+\frac{\pi}{2},\text{n}\in\text{z}$ or$\text{x}=2\text{n}\pi-\frac{\pi}{2},\text{n}\in\text{z}$
$\Rightarrow5\text{x}(4\text{n}+1)\frac{\pi}{2},\text{n}\in\text{z}$ or $\Rightarrow\text{x}(4\text{n}-1)\frac{\pi}{2}$
$\Rightarrow\text{x}(4\text{n}-1)\frac{\pi}{10},\text{n}\in\text{z}$or$\Rightarrow\text{x}(4\text{n}-1)\frac{\pi}{2}\text{n}\in\text{z}$

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