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Question Number 84259 by M±th+et£s last updated on 10/Mar/20

Commented by M±th+et£s last updated on 10/Mar/20

$$find\frac{y}{x}$$$$andz$$

Answered by mr W last updated on 10/Mar/20

Commented by mr W last updated on 10/Mar/20

$$BD=x=2R\cos \alpha$$$$AC=x=2R\tan \alpha$$$$\Rightarrow \cos \alpha =\tan \alpha =\frac{\sin \alpha }{\cos \alpha }$$$$\Rightarrow {\cos }^2\alpha =\sin \alpha$$$$\Rightarrow {\sin }^2\alpha +\sin \alpha -1=0$$$$\Rightarrow \sin \alpha =\frac{\sqrt{5}-1}{2}$$$$\Rightarrow \alpha ={\sin }^{-1}\frac{\sqrt{5}-1}{2}\approx 38.17°$$$$\Rightarrow z=\gamma =\frac{\pi }{2}-\alpha =\frac{\pi }{2}-{\sin }^{-1}\frac{\sqrt{5}-1}{2}$$$$\Rightarrow z={\cos }^{-1}\frac{\sqrt{5}-1}{2}\approx 51.83°$$$$\Rightarrow \frac{y}{x}=\cos z=\frac{\sqrt{5}-1}{2}$$

Commented by M±th+et£s last updated on 10/Mar/20

$$godblessyousir$$

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