Question

Find the exact value of cos285 using the fact that 285=330-45

Answer 1

$\bf \textit{Sum and Difference Identities} \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)\\\\ -------------------------------\\\\ cos(285^o) \\\\\\ cos(330^o-45^o)=cos(330^o)cos(45^o)+sin(330^o)sin(45^o) \\\\\\ cos(330^o-45^o)=\left( \frac{\sqrt{3}}{2} \right)\left( \frac{\sqrt{2}}{2} \right)~~+~~\left( \frac{1}{2} \right)\left( \frac{\sqrt{2}}{2} \right) \\\\\\ cos(330^o-45^o)=\cfrac{\sqrt{6}}{4}+\cfrac{\sqrt{2}}{4}\implies cos(330^o-45^o)=\cfrac{\sqrt{6}+\sqrt{2}}{4}$