Question

usando somas e diferença calcular cos de 165

Answer 1

$\cos 165^{\circ}=\cos \left(135^{\circ}+30^{\circ} \right )$


Utiliando a fórmula do cosseno da soma de dois arcos:

$\boxed{\begin{array}{c} \cos \left(a+b \right )=\cos a \cdot \cos b- \mathrm{sen\,}a \cdot \mathrm{sen\,}b \end{array}}$

e fazendo $a=135^{\circ}\;\text{ e }\;b=30^{\circ}$, temos:


$\cos 165^{\circ}=\cos \left(135^{\circ}+30^{\circ} \right )\\ \\ \cos 165^{\circ}=\cos 135^{\circ}\cdot \cos 30^{\circ}-\mathrm{sen\,}135^{\circ}\cdot \mathrm{sen\,}30^{\circ}\\ \\ \cos 165^{\circ}=\left(-\dfrac{\sqrt{2}}{2} \right )\cdot \dfrac{\sqrt{3}}{2}-\dfrac{\sqrt{2}}{2}\cdot \dfrac{1}{2}\\ \\ \cos 165^{\circ}=\dfrac{-\sqrt{2}\cdot \sqrt{3}}{2 \cdot 2}-\dfrac{\sqrt{2}\cdot 1}{2 \cdot 2}\\ \\ \cos 165^{\circ}=\dfrac{-\sqrt{2\cdot 3}}{4}-\dfrac{\sqrt{2}}{4}\\ \\ \cos 165^{\circ}=\dfrac{-\sqrt{6}}{4}-\dfrac{\sqrt{2}}{4}\\ \\ \boxed{\begin{array}{c} \cos 165^{\circ}=\dfrac{-\sqrt{6}-\sqrt{2}}{4} \end{array}}$