Question

qual e seno 15? tangente 15?

Answer 1

$sen15\º = sen(45\º-30\º) \\\\ sen(45\º-30\º) = sen45\º \cdot cos30\º - sen30\º \cdot cos 45\º \\\\ sen(45\º-30\º) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} \\\\ sen(45\º-30\º) = \frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4} \\\\ \boxed{\boxed{sen(15\º ) = \frac{\sqrt{6}-\sqrt{2}}{4}}}$

E com a tangente:

$tg15\º = tg(45\º-30\º) \\\\ tg(45\º-30\º) = \frac{tg45\º-tg30\º}{1+tg45\º \cdot tg30\º} \\\\ tg(45\º-30\º) = \frac{1-\frac{\sqrt{3}}{3}}{1+1 \cdot \frac{\sqrt{3}}{3}} \\\\ tg(45\º-30\º) = \frac{\frac{3-\sqrt{3}}{3}}{\frac{3+\sqrt{3}}{3}} \\\\ tg(45\º-30\º) = \frac{3 \cdot (3-\sqrt{3})}{3 \cdot (3+\sqrt{3})} \\\\ \boxed{\boxed{tg15 \º = \frac{3-\sqrt{3}}{3+\sqrt{3}}}}$