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Question Number 115685 by ZiYangLee last updated on 27/Sep/20

$$\mathrm{If}\alpha ,\beta ,\gamma ,\delta \mathrm{are}\mathrm{the}\mathrm{smallest}\mathrm{positive}$$$$\mathrm{angles}\mathrm{in}\mathrm{ascending}\mathrm{order}\mathrm{of}$$$$\mathrm{magnitude}\mathrm{which}\mathrm{have}\mathrm{their}\mathrm{sines}$$$$\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{positive}\mathrm{quantity}k,\mathrm{then}$$$$\mathrm{the}\mathrm{value}\mathrm{of}$$$$4\sin \frac{\alpha }{2}+3\sin \frac{\beta }{2}+2\sin \frac{\gamma }{2}+\sin \frac{\delta }{2}\mathrm{is}$$$$\mathrm{equal}\mathrm{to}$$

Answered by mr W last updated on 27/Sep/20

$$\sin \alpha =\sin \beta =\sin \gamma =\sin \delta =k>0$$$$\Rightarrow \alpha ={\sin }^{-1}kwith0<\alpha <\frac{\pi }{2}$$$$\Rightarrow \beta =\pi -\alpha$$$$\Rightarrow \gamma =2\pi +\alpha$$$$\Rightarrow \delta =3\pi -\alpha$$$$4\sin \frac{\alpha }{2}+3\sin \frac{\beta }{2}+2\sin \frac{\gamma }{2}+\sin \frac{\delta }{2}$$$$=4\sin \frac{\alpha }{2}+3\sin (\frac{\pi }{2}-\frac{\alpha }{2})+2\sin (\pi +\frac{\alpha }{2})+\sin (\frac{3\pi }{2}-\frac{\alpha }{2})$$$$=4\sin \frac{\alpha }{2}+3\cos \left(\frac{\alpha }{2}\right)-2\sin \left(\frac{\alpha }{2}\right)-\cos \left(\frac{\alpha }{2}\right)$$$$=2(\sin \frac{\alpha }{2}+\cos \frac{\alpha }{2})$$$$=2\sqrt{{(\sin \frac{\alpha }{2}+\cos \frac{\alpha }{2})}^2}$$$$=2\sqrt{1+\sin \alpha }$$$$=2\sqrt{1+k}$$

Commented by ZiYangLee last updated on 29/Sep/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}...$$

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