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Question Number 168277 by Eulerian last updated on 07/Apr/22

Prove that ((sin(((3π)/5)))/(sin(((4π)/5)))) = ((1+(√5))/2)

$$\:\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{5}}\right)}{\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)}\:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\: \\ $$

Commented by cortano1 last updated on 07/Apr/22

((sin 108°)/(sin 144°)) = ((sin (90°+18°))/(sin (180°−36°))) = ((cos 18°)/(sin 36°)) = ((cos 18°)/(2sin 18° cos 18°)) = (1/(2sin 18°)) = (1/(2((((√5)−1)/4)))) = (2/( (√5)−1))×(((√5)+1)/( (√5)+1)) = (((√5)+1)/2)

$$\:\:\frac{\mathrm{sin}\:\mathrm{108}°}{\mathrm{sin}\:\mathrm{144}°}\:=\:\frac{\mathrm{sin}\:\left(\mathrm{90}°+\mathrm{18}°\right)}{\mathrm{sin}\:\left(\mathrm{180}°−\mathrm{36}°\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{cos}\:\mathrm{18}°}{\mathrm{sin}\:\mathrm{36}°}\:=\:\frac{\mathrm{cos}\:\mathrm{18}°}{\mathrm{2sin}\:\mathrm{18}°\:\mathrm{cos}\:\mathrm{18}°} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2sin}\:\mathrm{18}°}\:=\:\frac{\mathrm{1}}{\mathrm{2}\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{5}}−\mathrm{1}}×\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\:\sqrt{\mathrm{5}}+\mathrm{1}}\:=\:\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{2}} \\ $$

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