Question

calcule os valores da tangente e do cosseno de pi/12.

Answer 1

$\tan( \frac{\pi}{12} ) = \tan(15) \\ = \tan(45 - 30) \\ = \frac{ \tan(45) - \tan(30) }{1 + \tan(45). \tan(30) }$

$\tan(15) = \frac{1- \frac{ \sqrt{3} }{3} }{1 + 1. \frac{ \sqrt{3} }{3} }$

$\tan(15) = \frac{ \frac{3 - \sqrt{3} }{3} }{ \frac{3 + \sqrt{3} }{3} } \\ \tan(15) = \frac{3 - \sqrt{3} }{3 + \sqrt{3} }$

$\tan(15) = \frac{(3 - \sqrt{3})(3 - \sqrt{3} )}{(3 + \sqrt{3} )(3 - \sqrt{3} )} \\ \tan(15) = \frac{9 - 6 \sqrt{3} + 3 }{9 - 3}$

$\tan(15) = \frac{12 - 6 \sqrt{3} }{6} \\ = \frac{6(2 - \sqrt{3})}{6} \\ \tan(15) = 2 - \sqrt{3}$

$\cos( \frac{\pi}{12} ) = \cos(15) = \cos(45 - 30)$

$\cos(45) \cos(30) + \sin(45) \sin(30) \\ = \frac{ \sqrt{2} }{2}. \frac{ \sqrt{3} }{2} + \frac{ \sqrt{2} }{2}. \frac{1}{2} = \frac{ \sqrt{6} + \sqrt{2} }{4}$