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Question Number 115685 by ZiYangLee last updated on 27/Sep/20
If α, β, γ, δ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of 4 sin (α/2)+3 sin (β/2)+2 sin (γ/2)+sin (δ/2) is equal to
$$\mathrm{If}\:\:\alpha,\:\beta,\:\gamma,\:\delta\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\: \\ $$$$\mathrm{angles}\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{order}\:\mathrm{of} \\ $$$$\mathrm{magnitude}\:\mathrm{which}\:\mathrm{have}\:\mathrm{their}\:\mathrm{sines}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{quantity}\:{k},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{4}\:\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{3}\:\mathrm{sin}\:\frac{\beta}{\mathrm{2}}+\mathrm{2}\:\mathrm{sin}\:\frac{\gamma}{\mathrm{2}}+\mathrm{sin}\:\frac{\delta}{\mathrm{2}}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$
Answered by mr W last updated on 27/Sep/20
sin α=sin β=sin γ=sin δ=k>0 ⇒α=sin^(−1) k with 0<α<(π/2) ⇒β=π−α ⇒γ=2π+α ⇒δ=3π−α 4 sin (α/2)+3 sin (β/2)+2 sin (γ/2)+sin (δ/2) =4 sin (α/2)+3 sin ((π/2)−(α/2))+2 sin (π+(α/2))+sin (((3π)/2)−(α/2)) =4 sin (α/2)+3 cos ((α/2))−2 sin ((α/2))−cos ((α/2)) =2(sin (α/2)+cos (α/2)) =2(√((sin (α/2)+cos (α/2))^2 )) =2(√(1+sin α)) =2(√(1+k))
$$\mathrm{sin}\:\alpha=\mathrm{sin}\:\beta=\mathrm{sin}\:\gamma=\mathrm{sin}\:\delta={k}>\mathrm{0} \\ $$$$\Rightarrow\alpha=\mathrm{sin}^{−\mathrm{1}} {k}\:{with}\:\mathrm{0}<\alpha<\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\beta=\pi−\alpha \\ $$$$\Rightarrow\gamma=\mathrm{2}\pi+\alpha \\ $$$$\Rightarrow\delta=\mathrm{3}\pi−\alpha \\ $$$$\mathrm{4}\:\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{3}\:\mathrm{sin}\:\frac{\beta}{\mathrm{2}}+\mathrm{2}\:\mathrm{sin}\:\frac{\gamma}{\mathrm{2}}+\mathrm{sin}\:\frac{\delta}{\mathrm{2}} \\ $$$$=\mathrm{4}\:\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{3}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\alpha}{\mathrm{2}}\right)+\mathrm{2}\:\mathrm{sin}\:\left(\pi+\frac{\alpha}{\mathrm{2}}\right)+\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}−\frac{\alpha}{\mathrm{2}}\right) \\ $$$$=\mathrm{4}\:\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{3}\:\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{2}\:\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right) \\ $$$$=\mathrm{2}\left(\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{cos}\:\frac{\alpha}{\mathrm{2}}\right) \\ $$$$=\mathrm{2}\sqrt{\left(\mathrm{sin}\:\frac{\alpha}{\mathrm{2}}+\mathrm{cos}\:\frac{\alpha}{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$=\mathrm{2}\sqrt{\mathrm{1}+\mathrm{sin}\:\alpha} \\ $$$$=\mathrm{2}\sqrt{\mathrm{1}+{k}} \\ $$
Commented by ZiYangLee last updated on 29/Sep/20
thank you sir...
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}… \\ $$

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