Question

Cálculo da secante de 15°

Answer 1

Olá, Lidesrosies.

$\cos 15\º = \cos (45\º -30\º) = \cos 45\º \cos 30\º + \sin 45\º \sin 30\º = \\\\ =\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac{1}{2}=\frac{\sqrt6}4+\frac{\sqrt2}4=\frac{\sqrt6+\sqrt2}4\\\\ \sec15\º=\frac1{\cos15\º}=\frac4{\sqrt6+\sqrt2}=\frac4{\sqrt6+\sqrt2}\cdot\frac{\sqrt6-\sqrt2}{\sqrt6-\sqrt2}=\frac{4(\sqrt6-\sqrt2)}{6-2}\Rightarrow\\\\ \boxed{\sec15\º=\sqrt6-\sqrt2}$