Question Number 192282 by universe last updated on 14/May/23
$$\:\:\:{find}\:{the}\:{value}\:{of} \\ $$$$\:\:\:\:\mathrm{tan}\:\frac{\pi}{\mathrm{9}}\:+\:\mathrm{4sin}\:\frac{\pi}{\mathrm{9}}\:=\:? \\ $$
Answered by som(math1967) last updated on 14/May/23
$${tan}\mathrm{20}+\mathrm{4}{sin}\mathrm{20} \\ $$$$\frac{{sin}\mathrm{20}+\mathrm{4}{sin}\mathrm{20}{cos}\mathrm{20}}{{cos}\mathrm{20}} \\ $$$$=\frac{{sin}\mathrm{20}+\mathrm{2}{sin}\mathrm{40}}{{cos}\mathrm{20}} \\ $$$$=\frac{{sin}\mathrm{20}+{sin}\mathrm{40}+{sin}\mathrm{40}}{{cos}\mathrm{20}} \\ $$$$=\frac{\mathrm{2}{sin}\mathrm{30}{cos}\mathrm{10}+{cos}\mathrm{50}}{{cos}\mathrm{20}} \\ $$$$=\frac{{cos}\mathrm{10}+{cos}\mathrm{50}}{{cos}\mathrm{20}} \\ $$$$=\frac{\mathrm{2}{cos}\mathrm{30}{cos}\mathrm{20}}{{cos}\mathrm{20}} \\ $$$$=\sqrt{\mathrm{3}} \\ $$