Question

Quanto vale tg $\pi \\ 18$ +tg $\pi \\ 9$ / 1- tg $\pi \\ 18$ . tg $\pi \\ 9$ 

Answer 1

Tem uma fórmula na trigonometria que diz o seguinte:

$\boxed{\mathrm{tg}\hspace{0,2mm}(a+b)=\frac{\mathrm{tg}\hspace{0,2mm}a+\mathrm{tg}\hspace{0,2mm}b}{1-\mathrm{tg}\hspace{0,2mm}a.\mathrm{tg}\hspace{0,2mm}b}}$

Fazendo $a=\frac{\pi}{18}$ e $b=\frac{\pi}{9}$ teremos:

$\mathrm{tg}\hspace{0,2mm}\left(\frac{\pi}{18}+\frac{\pi}{9}\right)=\frac{\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}+\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}{1-\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}.\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}$

$\mathrm{tg}\hspace{0,2mm}\left(\frac{3\pi}{18}\right)=\frac{\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}+\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}{1-\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}.\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}$

$\mathrm{tg}\hspace{0,2mm}\left(\frac{\pi}{6}\right)=\frac{\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}+\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}{1-\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}.\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}$

$\boxed{\boxed{\frac{\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}+\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}{1-\mathrm{tg}\hspace{0,2mm}\frac{\pi}{18}.\mathrm{tg}\hspace{0,2mm}\frac{\pi}{9}}=\frac{\sqrt3}{3}}}$