Question

resolva :

cos105° - sen 75 = ?

Answer 1

105 - 90 = 15

cos 15 = cos (60-45) = cos 60. cos 45 + sen 60. sen 45 = 1/2.\/2/2 + \/3/2.\/2/2 = 

\/2/4 + \/6/4 = (\/2+\/6)/4 --> cos 15º

 

cos 105 = - cos 15 = - (\/2+\/6)/4

 

sen 75 = sen (30+45) = sen 30. cos 45 + sen 45 . cos 30 = 1/2.\/2/2 + \/2/2 . \/3/2 = 

\/2/4 + \/6/4 = (\/2+\/6)/4

 

cos 105 - sen 75 = -(\/2+\/6)/4 + (\/2+\/6)/4 = 0

Answer 2

$<var>cos105 - sen75</var>$

 

$<var>cos(60+45) - sen(30+45)</var>$

 

Aplicando a propriedade:

$<var>cos(a+b) \\ cosa*cosb-sena*senb \\\\ sen(a+b) \\ sena*cosb+senb*cosa</var>$

 

 

$<var>cos(60+45) - sen(30+45) \\ cos60*cos45-sen60*sen45 - sen30*cos45+sen45*cos30 \\ \frac{1}{2}*\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}*\frac{\sqrt{2}}{2}-\frac{1}{2}*\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}*\frac{\sqrt{3}}{2} \\ \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}</var>$

 

Anula-se tudo.

 

$<var>cos105-sen75 = \boxed{0}</var>$