Question

Calcule o valor de cos 75º- sen 105º.

Answer 1

$cos (a + b) = cos a * cos b - sen a * sen b\\ cos75=cos(45+30)\\ cos75=cos45*cos30-sen45*sen30\\ cos75=(\sqrt{2}/2*\sqrt{3}/2)-(\sqrt{2}/2*1/2)\\ cos75=(\sqrt{6}/4-\sqrt{2}/4)\\ cos75=(\sqrt{6}-\sqrt{2})/4$


$sen (a + b) = sen a * cos b + cos a * sen b\\ sen105=sen(60+45)\\ sen105=sen60*cos45+cos60*sen45\\ sen105=(\sqrt{3}/2*\sqrt{2}/2)+(1/2*\sqrt{2}/2)\\ sen105=(\sqrt{6}/4)+(\sqrt{2}/4)\\ sen105=(\sqrt{6}+\sqrt{2})/4$


$cos75-sen105\\ ((\sqrt{6}-\sqrt{2})/4)-((\sqrt{6}+\sqrt{2})/4)\\ (\sqrt{6}-\sqrt{6}-\sqrt{2}-\sqrt{2})/4\\ -2\sqrt{2}/4\\\\ cos75-sen105=-\sqrt{2}/2$