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Question Number 128285 by john_santu last updated on 06/Jan/21

cos ((π/7))−cos (((2π)/7))+cos (((3π)/7)) =?

$$\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)−\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:=? \\ $$

Answered by liberty last updated on 06/Jan/21

let ϕ = cos ((π/7))−cos (((2π)/7))+cos (((3π)/7)) ϕ = cos ((π/7))+cos (((3π)/7))+cos (((5π)/7)) ⇒2ϕsin t = 2sin t cos t + 2sin t cos 3t+2sin tcos 5t where t = (π/7) ⇒2ϕsin t = sin 2t + sin 4t−sin 2t+sin 6t−sin 4t ⇒2ϕsin t = sin 6t ⇒ϕ = ((sin (((6π)/7)))/(2 sin ((π/7)))) = (1/2)

$$\:\mathrm{let}\:\varphi\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)−\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right) \\ $$$$\:\varphi\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{5}\pi}{\mathrm{7}}\right) \\ $$$$\:\Rightarrow\mathrm{2}\varphi\mathrm{sin}\:\mathrm{t}\:=\:\mathrm{2sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{t}\:+\:\mathrm{2sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{3t}+\mathrm{2sin}\:\mathrm{tcos}\:\mathrm{5t} \\ $$$$\:\mathrm{where}\:\mathrm{t}\:=\:\frac{\pi}{\mathrm{7}} \\ $$$$\Rightarrow\mathrm{2}\varphi\mathrm{sin}\:\mathrm{t}\:=\:\mathrm{sin}\:\mathrm{2t}\:+\:\mathrm{sin}\:\mathrm{4t}−\mathrm{sin}\:\mathrm{2t}+\mathrm{sin}\:\mathrm{6t}−\mathrm{sin}\:\mathrm{4t} \\ $$$$\Rightarrow\mathrm{2}\varphi\mathrm{sin}\:\mathrm{t}\:=\:\mathrm{sin}\:\mathrm{6t}\: \\ $$$$\Rightarrow\varphi\:=\:\frac{\mathrm{sin}\:\left(\frac{\mathrm{6}\pi}{\mathrm{7}}\right)}{\mathrm{2}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{7}}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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