Question Number 18584 by 433 last updated on 25/Jul/17
$${Prove} \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{10}}\right)=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$
Answered by Tinkutara last updated on 25/Jul/17
![sin 54° = cos 36° = 1 − 2 sin^2 18° = 1 − ((3 − (√5))/4) = (((√5) + 1)/4) [∵ sin 18° = (((√5) − 1)/4)]](https://www.tinkutara.com/question/Q18585.png)
$$\mathrm{sin}\:\mathrm{54}°\:=\:\mathrm{cos}\:\mathrm{36}°\:=\:\mathrm{1}\:−\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{18}° \\ $$$$=\:\mathrm{1}\:−\:\frac{\mathrm{3}\:−\:\sqrt{\mathrm{5}}}{\mathrm{4}}\:=\:\frac{\sqrt{\mathrm{5}}\:+\:\mathrm{1}}{\mathrm{4}}\:\left[\because\:\mathrm{sin}\:\mathrm{18}°\:=\:\frac{\sqrt{\mathrm{5}}\:−\:\mathrm{1}}{\mathrm{4}}\right] \\ $$