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Question Number 162776 by john_santu last updated on 01/Jan/22

$$\frac{\sin x}{1-\cos x}=\frac{1}{2}$$$$x=?$$

Answered by bobhans last updated on 01/Jan/22

$$2\sin \mathrm{x}+\cos \mathrm{x}=1$$$$\sqrt{5}(\frac{2}{\sqrt{5}}\sin \mathrm{x}+\frac{1}{\sqrt{5}}\cos \mathrm{x})=1$$$$\cos (\mathrm{x}-\alpha )=\frac{1}{\sqrt{5}};\alpha =\arctan \left(2\right)$$$$\mathrm{x}=\pm \arccos \left(\frac{1}{\sqrt{5}}\right)+\arctan \left(2\right)+2\mathrm{k}\pi ,\mathrm{k}\in \mathbb{Z}$$

Answered by Rasheed.Sindhi last updated on 01/Jan/22

$$\frac{\sin x}{1-\cos x}=\frac{1}{2};x=?$$$$2\sin x=1-\cos x$$$${4\sin }^{2}x=1+{\cos }^{2}x-2\cos x$$$$4(1-{\cos }^{2}x)=1+{\cos }^{2}x-2\cos x$$$${5\cos }^{2}x-2\cos x-3=0$$$${5\cos }^{2}x-5\cos x+3\cos x-3=0$$$$5\cos x(\cos x-1)+3(\cos x-1)=0$$$$\underset{\underset{\left(\mathcal{E}xtraneous\right)}{x=2n\pi }}{\cos x=1}\mid \underset{\underset{\underset{x\approx 126.87+2n\pi }{}}{\stackrel{}{x={\cos }^{-1}\left(\frac{-3}{5}\right)}}+2n\pi }{\cos x=-\frac{3}{5}}$$

Answered by Berlindo last updated on 02/Jan/22

$$\frac{2\sin \left(\mathrm{x}/2\right)\cdot \cos \left(\mathrm{x}/2\right)}{2\sin {}^2\left(\mathrm{x}/2\right)}=\frac{1}{2}$$$$\cot \left(\mathrm{x}/2\right)=1/2\iff \tan \left(\mathrm{x}/2\right)=2$$$$\mathrm{x}/2={\tan }^{-1}2+\pi \mathrm{n},\mathrm{n}\in \mathbb{Z}$$$$\mathrm{x}={2\tan }^{-1}2+2\pi \mathrm{n},\mathrm{n}\in \mathbb{Z}$$

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