Respostas

2014-07-19T16:30:30-03:00
Cos 105° =
cos (60 + 45) =
cos60.cos45 - sen60.sen45 =
1/2 . √2 /2 - √3 /2 . √2 /2 =
√2 /4 - √6 / 4 ]=
(√2 - √6) / 4 

tg 15 = tg (45 - 30)

tg 15 =  \frac{tg45 - tg 30}{1 + tg45.tg30} =  \frac{1 -  \frac{ \sqrt{3} }{3} }{1 + 1 .  \frac{ \sqrt{3} }{3} }  =   \frac{ \frac{3- \sqrt{3} }{3} }{ \frac{3 +  \sqrt{3} }{3} }  =  \frac{3- \sqrt{3} }{3} .  \frac{3}{3 +  \sqrt{3} }   =  \frac{3 -  \sqrt{3} }{3 +  \sqrt{3} }  . \frac{3 -  \sqrt{3} }{3 -  \sqrt{3} } =   \\ \frac{9 - 6 \sqrt{3}+3 }{9 - 3} =  \frac{12 - 6 \sqrt{3} }{6} =  \frac{6 . (2 -  \sqrt{3} )}{6} = 2 -  \sqrt{3}

Logo tg 15 = 2 - √3