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Question Number 133851 by physicstutes last updated on 24/Feb/21

A circle of radius r is such that it subtends an  angle of α at its centre. A chord cuts the circle such  that it divides the circle into two segments of areas in  the ratio 1:5. show that     sin α = α−(π/3)

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:{r}\:\mathrm{is}\:\mathrm{such}\:\mathrm{that}\:\mathrm{it}\:\mathrm{subtends}\:\mathrm{an} \\ $$$$\mathrm{angle}\:\mathrm{of}\:\alpha\:\mathrm{at}\:\mathrm{its}\:\mathrm{centre}.\:\mathrm{A}\:\mathrm{chord}\:\mathrm{cuts}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{it}\:\mathrm{divides}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{into}\:\mathrm{two}\:\mathrm{segments}\:\mathrm{of}\:\mathrm{areas}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{ratio}\:\mathrm{1}:\mathrm{5}.\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\mathrm{sin}\:\alpha\:=\:\alpha−\frac{\pi}{\mathrm{3}} \\ $$

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