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Question Number 196084 by Mingma last updated on 17/Aug/23

Answered by mr W last updated on 17/Aug/23

Commented by mr W last updated on 17/Aug/23

$$R^2=a^2+{\left(\frac{a}{\cos \theta }\right)}^2-2a\left(\frac{a}{\cos \theta }\right)\cos (\frac{\pi }{2}+\theta )$$$$R^2=({\tan }^2\theta +2\tan \theta +2)a^2$$$$A_{green}=\frac{R^2\sin \theta \cos \theta }{2}=\frac{\sin 2\theta ({\tan }^2\theta +2\tan \theta +2)a^2}{4}=a^2$$$$\sin 2\theta ({\tan }^2\theta +2\tan \theta +2)=4$$$$\frac{2\tan \theta }{1+{\tan }^2\theta }({\tan }^2\theta +2\tan \theta +2)=4$$$${\tan }^3\theta +2\tan \theta -2=0$$$$\Rightarrow \tan \theta =\sqrt[3]{\frac{\sqrt{105}}{9}+1}-\sqrt[3]{\frac{\sqrt{105}}{9}-1}$$$$\Rightarrow \theta ={\tan }^{-1}(\sqrt[3]{\frac{\sqrt{105}}{9}+1}-\sqrt[3]{\frac{\sqrt{105}}{9}-1})$$$$=37.63°$$

Commented by Mingma last updated on 18/Aug/23

Perfect sir!

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