Question

Calcule o valor numérico da expressão:
(sen 30° - sec 120°).(cossec 150° - cotg330°)/ (sen 300° + tg 60° . cotg 225°)

Answer 1

Para facilitar vamos primeiro passar todos angulos para o primeiro quadrante:

sec(120°) = -sec(60°)

cossec(150°) = sen(30°)

cotg(330°) = -cotg(30°)

sen(300°) = -sen(60°)

cotg(225°) = cotg(45°)

Agora basta substituir os valores dos senos, cossenos, tgs ...

Lembrando cotg(x) = cos(x) / sen(x) ; cossec(x) = 1/sen(x) ; sec(x) = 1/cos(x)

$\frac{\left(sen30^\circ-(-sec60^\circ)\right).\left(sen30^\circ-(-cotg30^\circ\right)}{\left(-sen60^\circ+tg60^\circ\;.\;cotg45^\circ\right)} \\\\\frac{\left(sen30^\circ+sec60^\circ\right).\left(sen30^\circ+cotg30^\circ\right)}{\left(-sen60^\circ+tg60^\circ\;.\;cotg45^\circ\right)}\\\\\frac{\left(\frac{1}{2}+\frac{2}{1}\right).\left(\frac{1}{2}+\frac{3}{\sqrt{3}}\right)}{\left(-\frac{2}{\sqrt{3}}+\sqrt{3}\;.\;1\right)}$

$\frac{\left(\frac{5}{2}\right).\left(\frac{\sqrt{3}+6}{2\sqrt{3}}\right)}{\left(\frac{-2+\sqrt{3}^2}{\sqrt{3}}\right)}\\\\\\\frac{\left(\frac{5\sqrt{3}+30}{4\sqrt{3}}\right)}{\left(\frac{-2+3}{\sqrt{3}}\right)}\\\\\\\left(\frac{5\sqrt{3}+30}{4\sqrt{3}}\right).\left(\frac{\sqrt{3}}{1}\right)\\\\\\\frac{5\sqrt{3}+30}{4}$