Question

O valor numérico da expressão:$\frac{sen\frac{\pi }{2}+ cos 240º - [tg (-750º)^{2} }{(sec 1200º) (cossec \frac{9\pi }{4}) + (cotg \frac{5\pi }{6} )^{2} }$

Answer 1

Oie, Td Bom?!

$= \frac{ \sin( \frac{\pi}{2} ) + \cos(240°) - \tan( - 750°) {}^{2} }{ \sec(1200°) \csc( \frac{9\pi}{4} ) + \cot( \frac{5\pi}{6} ) {}^{2} }$

➭Calculando por partes:

1.

$= \sin( \frac{\pi}{2} )$

$= 1$

2.

$= \cos(240°)$

$= \cos(240° \: . \: \frac{\pi}{180°} )$

$= \cos( \frac{4\pi}{3} )$

3.

$= - \tan( - 750°)$

$= - \tan( - 750° \: . \: \frac{\pi}{180°} )$

$= - \tan( - \frac{25\pi}{6} )$

4.

$= \sec(1200°)$

$= \sec(1200° \: . \: \frac{\pi}{180°} )$

$= \sec( \frac{20\pi}{3} )$

5.

$= \csc( \frac{9\pi}{4} )$

$= \csc( \frac{\pi}{4} + 2\pi)$

$= \csc( \frac{\pi}{4} )$

$= \sqrt{2}$

6.

$= \cot( \frac{5\pi}{3} )$

$= - \sqrt{3}$

• Continuando:

$= \frac{1 + \cos( \frac{4\pi}{3} ) - \tan( - \frac{25\pi}{6} ) {}^{2} }{ \sec( \frac{20\pi}{3} ) \: . \: \sqrt{2} + ( - \sqrt{3} ) {}^{2} }$

➭Calculando por partes:

1.

$= \cos( \frac{4\pi}{3} )$

$= - \frac{1}{2}$

2.

$= \tan( - \frac{25\pi}{6} )$

$= - \tan( \frac{25\pi}{6} )$

$= - \tan( \frac{\pi}{6} + 4\pi)$

$= - \tan( \frac{\pi}{6} )$

$= - \frac{ \sqrt{3} }{3}$

3.

$= \sec( \frac{20\pi}{3} )$

$= \sec( \frac{2\pi}{3} + 2 \: . \: 3\pi)$

$= \sec( \frac{2\pi}{3} )$

$= - 2$

4.

$= ( - \sqrt{3} ) {}^{2}$

$= \sqrt{3} {}^{2}$

• Continuando:

$= \frac{1 - \frac{1}{2} - ( - \frac{ \sqrt{3} }{3} ) {}^{2} }{ - 2 \sqrt{2} + \sqrt{3} {}^{2} }$

$= \frac{1 - \frac{1}{2} - \frac{3}{9} }{ - 2 \sqrt{2} + 3}$

$= \frac{1 - \frac{1}{2} - \frac{1}{3} }{ - 2 \sqrt{2} + 3}$

$= \frac{ \frac{6 - 3 - 2}{6} }{ - 2 \sqrt{2} + 3 }$

$= \frac{ \frac{1}{6} }{ - 2 \sqrt{2} + 3}$

$= \frac{1}{6} \div ( - 2\sqrt{2} + 3)$

$= \frac{1}{6} \: . \: \frac{1}{ - 2 \sqrt{2} + 3}$

$= \frac{1 \: . \: 1}{6( - 2\sqrt{2} + 3)}$

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

➭Racionalizando:

$= \frac{1}{6( - 2 \sqrt{2} + 3) } \: .\: \frac{ - 2 \sqrt{2} - 3 }{ - 2 \sqrt{2} - 3}$

$= \frac{1( - 2 \sqrt{2} - 3)}{6( - 2 \sqrt{2} + 3) \: . \: ( - 2 \sqrt{2} - 3) }$

$= \frac{ - 2 \sqrt{2} - 3}{6(4 \: . \: 2 - 9)}$

$= \frac{ - 2 \sqrt{2} - 3}{6(8 - 9)}$

$= \frac{ - 2 \sqrt{2} - 3}{6 \: . \: ( - 1)}$

$= \frac{ - 2 \sqrt{2} - 3}{ - 6}$

$= - \frac{ - 2 \sqrt{2} - 3 }{6}$

$= - \frac{ - (2 \sqrt{2} + 3) }{6}$

$= \frac{2 \sqrt{2} + 3}{6}$

$≈0,971405...$

Att. Makaveli1996