Question

Determine o valor da expressão:

$( \sin8\pi - \cos5\pi) \\ \: \: \: \: \: \:\: \: \:3 \\ - - - - - - - - \\ \ \: \: \: \: tan13\pi \\ \: \: \: \: \: \: \: \: \: \: 6$

Answer 1

$\sin( \frac{8\pi}{3} ) = \sin(2\pi + \frac{2\pi}{3} ) = \sin( \frac{2\pi}{3} ) = \frac{ \sqrt{3} }{2}$

$\cos( 5\pi ) = \cos(4\pi + \pi) = \cos(\pi) = - 1$

$\tan( \frac{13\pi}{6} ) = \tan(2\pi + \frac{\pi}{6} ) = \frac{ \sqrt{3} }{3}$

Logo,

$\frac{ \sin( \frac{8\pi}{3} ) - cos (5\pi) }{ \tan( \frac{13\pi}{6} ) } = \frac{ \frac{ \sqrt{3} }{2} + 1 }{ \frac{ \sqrt{3} }{3} } = \frac{2 +\sqrt{3} }{2} \times \frac{3}{ \sqrt{3} } = \\ = \frac{3}{2} \times \frac{2 + \sqrt{3} }{ \sqrt{3} } = \frac{6 +3 \sqrt{3} }{2 \sqrt{3} } = \\ = \frac{6 \sqrt{3} + 9 }{6} = \frac {2\sqrt{3}+ 3}{2}$

Answer 2

É mais simples resolver por partes e depois substituir.

$Sen(\frac{8\pi}{3}) = Sen(\frac{6\pi}{3} + \frac{2\pi}{3}) = Sen(\frac{2\pi}{3})$

2 * 180/3

120 ⇒ 180 -120 = 60

Sen(60) = √3/2

$Cos(5\pi) = Cos(0) = -1$

$Tan(\frac{13\pi}{6}) = Tan(\frac{12\pi}{6} + \frac{\pi}{6}) = Tan(\frac{\pi}{6})$

180/6 = 30

Tan(30) = √3/3

→ Substituindo

$\frac{\frac{\sqrt{3}}{2} -(-1)}{\frac{\sqrt{3}}{3}} = \frac{2 + \sqrt{3}}{2} * \frac{3}{\sqrt{3}} = \frac{6 + 3\sqrt{3}}{2\sqrt{3}} * \frac{2\sqrt{3}}{2\sqrt{3}} = \frac{12\sqrt{3} + 18}{12} = \frac{2\sqrt{3} + 3}{2}$