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Question Number 188463 by emilagazade last updated on 01/Mar/23

Answered by mr W last updated on 01/Mar/23

$$\frac{AO}{CO}=\frac{\sin 30}{\sin 70}$$$$\frac{BO}{AO}=\frac{\sin 40}{\sin \alpha }$$$$\frac{CO}{BO}=\frac{\sin (20-\alpha )}{\sin 20}$$$$\left(i\right)\times \left(ii\right)\times \left(iii\right):$$$$1=\frac{\sin 30\times \sin 40\times \sin (20-\alpha )}{\sin 70\times \sin \alpha \times \sin 20}$$$$\frac{\sin 20}{\tan \alpha }-\cos 20=\frac{\sin 70\times \sin 20}{\sin 30\times \sin 40}$$$$\tan \alpha =\frac{1}{\frac{\sin 70}{\sin 30\times \sin 40}+\frac{1}{\tan 20}}$$$$\Rightarrow \alpha ={\tan }^{-1}\left(\frac{1}{\frac{\sin 70}{\sin 30\times \sin 40}+\frac{1}{\tan 20}}\right)=10°$$

Commented by universe last updated on 02/Mar/23

$$\sin 70\times \sin \alpha \times \sin 20=\sin 30\times \sin 40\times \sin (20-\alpha )$$$$\sin 70\times \sin \alpha \times \cancel{\sin 20}=\frac{1}{2}2\cancel{\sin 20}\times \cos 20\times \sin (20-\alpha )$$$$\sin 70\times \sin \alpha =\sin 70\times \sin (20-\alpha )$$$$\sin \alpha =\sin (20-\alpha )$$$$\alpha =20-\alpha$$$$2\alpha =20$$$$\alpha =10°$$

Commented by mr W last updated on 02/Mar/23

$$great!$$

Commented by emilagazade last updated on 02/Mar/23

$$tanksalot$$

Commented by emilagazade last updated on 02/Mar/23

$$thankyouSir$$

Answered by HeferH last updated on 02/Mar/23

Commented by HeferH last updated on 02/Mar/23

$$20°=2\alpha$$$$\alpha =10°$$

Commented by mr W last updated on 02/Mar/23

��

Commented by emilagazade last updated on 02/Mar/23

$$Nice,thankyou$$

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