Question

Sabendo que sen 2β = 5/4 , calcule tg β + cotg β

Answer 1

Resposta:

$\boxed{tg\beta +cotg\beta = \frac{8}{5}}}$

Explicação passo-a-passo:

Vamos entender o que é tg e cotg

$tg\beta + cotg\beta \\\\\frac{sen\beta }{cos\beta } +\frac{cos\beta }{sen\beta } \\\\\frac{sen^{2}\beta+cos^{2} \beta   }{cos\beta sen\beta  } \\\\ \frac{1 }{cos\beta sen\beta  }$

Agora vamos manipular sen 2β = 5/4

$sen2\beta \\sen(\beta +\beta )\\sen\beta cos\beta +sen\beta cos\beta \\\\2sen\beta cos=\frac{5}{4} \\\\sen\beta cos=\frac{\frac{5}{4} }{2}\\ sen\beta cos=\frac{5}{4}.\frac{2}{1}\\  sen\beta cos=\frac{5}{8}$

Agora vamos substituir esse valor no primeiro desenvolvimento que fizemos:

$\frac{1}{sen\beta cos\beta  }=\frac{1}{\frac{5}{8} } \\\\\frac{1}{sen\beta cos\beta  }=1.\frac{8}{5}\\\frac{1}{sen\beta cos\beta  }=\frac{8}{5} \\\\\boxed{tg\beta +cotg\beta = \frac{8}{5}}}$