Question

Calcule cotg 35pi/6, sec 37pi/6, cossec 37pi/6

Answer 1

Lembrando, 2π equivale a 360°, uma volta completa.

a)

$~\,2\pi~_{-----}~360^\circ\\\frac{35}{6}\pi~_{-----}~~x\\\\\\Multiplicando~Cruzado\\\\\\2\pi~.~x~=~\frac{35}{6}\pi~.~360^\circ\\\\\\x~=~\frac{35\pi~.~360^\circ}{6~.~2\pi}\\\\\\\boxed{x~=~1050^\circ}$

Reduzindo para a primeira determinação positiva:

$1050^\circ~=~2~.~360^\circ~+~\boxed{330^\circ}$

Calculando a cotangente:

$cotg(330^\circ)~=~\frac{cos(330^\circ)}{sen(330^\circ)}~=~\frac{cos(360^\circ-330^\circ)}{-sen(360^\circ-330^\circ)}~=~\frac{cos(30^\circ)}{-sen(30^\circ)}~=~\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}~=~\boxed{-\sqrt{3}}$

b)

$~\,2\pi~_{-----}~360^\circ\\\frac{37}{6}\pi~_{-----}~~x\\\\\\Multiplicando~Cruzado\\\\\\2\pi~.~x~=~\frac{37}{6}\pi~.~360^\circ\\\\\\x~=~\frac{37\pi~.~360^\circ}{6~.~2\pi}\\\\\\\boxed{x~=~1110^\circ}$

Reduzindo para a primeira determinação positiva:

$1110^\circ~=~3~.~360^\circ~+~\boxed{30^\circ}$

Calculando a secante:

$sec(30^\circ)~=~\frac{1}{cos(30^\circ)}~=~\frac{1}{\frac{\sqrt{3}}{2}}~=~\frac{2}{\sqrt{3}}~=~\boxed{\frac{2\sqrt{3}}{3}}$

c)

$~\,2\pi~_{-----}~360^\circ\\\frac{37}{6}\pi~_{-----}~~x\\\\\\Multiplicando~Cruzado\\\\\\2\pi~.~x~=~\frac{37}{6}\pi~.~360^\circ\\\\\\x~=~\frac{37\pi~.~360^\circ}{6~.~2\pi}\\\\\\\boxed{x~=~1110^\circ}$

Reduzindo para a primeira determinação positiva:

$1110^\circ~=~3~.~360^\circ~+~\boxed{30^\circ}$

Calculando a cossecante:

$cossec(30^\circ)~=~\frac{1}{sen(30^\circ)}~=~\frac{1}{\frac{1}{2}}~=~\boxed{2}$