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If-0-lt-lt-pi-and-they-satisfy-cos-cos-cos-3-2-1-2-2pi-3-3-2-4-2-

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Question Number 18062 by Tinkutara last updated on 14/Jul/17
If 0 < α, β < π and they satisfy cos α + cos β − cos (α + β) = (3/2) (1) α = β (2) α + β = ((2π)/3) (3) α = 2β (4) β = 2α
$$\mathrm{If}\:\mathrm{0}\:<\:\alpha,\:\beta\:<\:\pi\:\mathrm{and}\:\mathrm{they}\:\mathrm{satisfy} \\ $$$$\mathrm{cos}\:\alpha\:+\:\mathrm{cos}\:\beta\:−\:\mathrm{cos}\:\left(\alpha\:+\:\beta\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\:\alpha\:=\:\beta \\ $$$$\left(\mathrm{2}\right)\:\alpha\:+\:\beta\:=\:\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\left(\mathrm{3}\right)\:\alpha\:=\:\mathrm{2}\beta \\ $$$$\left(\mathrm{4}\right)\:\beta\:=\:\mathrm{2}\alpha \\ $$
Answered by Tinkutara last updated on 16/Jul/17
2 cos ((α + β)/2) cos ((α − β)/2) − 2 cos^2 ((α + β)/2) = (1/2) 4 cos^2 ((α + β)/2) − 4 cos ((α + β)/2) cos ((α − β)/2) + 1 = 0 (2 cos ((α + β)/2) − cos ((α − β)/2))^2 + sin^2 ((α − β)/2) = 0 ⇒ α = β = (π/3)
$$\mathrm{2}\:\mathrm{cos}\:\frac{\alpha\:+\:\beta}{\mathrm{2}}\:\mathrm{cos}\:\frac{\alpha\:−\:\beta}{\mathrm{2}}\:−\:\mathrm{2}\:\mathrm{cos}^{\mathrm{2}} \:\frac{\alpha\:+\:\beta}{\mathrm{2}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{4}\:\mathrm{cos}^{\mathrm{2}} \:\frac{\alpha\:+\:\beta}{\mathrm{2}}\:−\:\mathrm{4}\:\mathrm{cos}\:\frac{\alpha\:+\:\beta}{\mathrm{2}}\:\mathrm{cos}\:\frac{\alpha\:−\:\beta}{\mathrm{2}}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{2}\:\mathrm{cos}\:\frac{\alpha\:+\:\beta}{\mathrm{2}}\:−\:\mathrm{cos}\:\frac{\alpha\:−\:\beta}{\mathrm{2}}\right)^{\mathrm{2}} \:+\:\mathrm{sin}^{\mathrm{2}} \:\frac{\alpha\:−\:\beta}{\mathrm{2}}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\alpha\:=\:\beta\:=\:\frac{\pi}{\mathrm{3}} \\ $$

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