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Question Number 199730 by Rupesh123 last updated on 08/Nov/23

Answered by mr W last updated on 10/Nov/23

Commented by mr W last updated on 10/Nov/23

$$\frac{x}{\sin \theta }=\frac{1}{\sin 18°}=\frac{y}{\sin (\theta -18°)}$$$$\frac{y+1}{\sin (\theta +18°)}=\frac{1}{\sin 18°}$$$$y=\frac{\sin (\theta -18°)}{\sin 18°}=\frac{\sin (\theta +18°)}{\sin 18°}-1$$$$\frac{\sin (\theta +18°)-\sin (\theta -18°)}{\sin 18°}=1$$$$\Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \sin \theta =\frac{\sqrt{3}}{2}$$$$x=\frac{\sin \theta }{\sin 18°}=\frac{\sqrt{3}}{2\sin 18°}=\frac{4\sqrt{3}}{2(\sqrt{5}-1)}$$$$=\frac{\sqrt{15}+\sqrt{3}}{2}✓$$

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