Question Number 91919 by zainal tanjung last updated on 03/May/20
$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{sin}\:\mathrm{12}°\:\mathrm{sin}\:\mathrm{48}°\:\mathrm{sin}\:\mathrm{54}°\:\:\mathrm{is} \\ $$
Commented by Tony Lin last updated on 03/May/20
![sin12sin48sin54 (in deg) =(1/2)[cos(36)−cos(60)]sin54 =((sin^2 54)/2)−((sin54)/4) =((((((√5)−1)/4))^2 )/2)−((((√5)−1)/4)/4) =(1/8)](Q91924.png)
$${sin}\mathrm{12}{sin}\mathrm{48}{sin}\mathrm{54}\:\:\:\:\left({in}\:{deg}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[{cos}\left(\mathrm{36}\right)−{cos}\left(\mathrm{60}\right)\right]{sin}\mathrm{54} \\ $$$$=\frac{{sin}^{\mathrm{2}} \mathrm{54}}{\mathrm{2}}−\frac{{sin}\mathrm{54}}{\mathrm{4}} \\ $$$$=\frac{\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{2}} }{\mathrm{2}}−\frac{\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}} \\ $$