Question

Sabendo que cossec x = 2√3/3 e ₢/2 < x < ₢. calcular tg x/2

Answer 1

$\mathrm{cossec\,}x=\dfrac{2\sqrt{3}}{3}\\ \\ \dfrac{1}{\mathrm{sen\,}x}=\dfrac{2\sqrt{3}}{3}\\ \\ \mathrm{sen\,}x=\dfrac{3}{2\sqrt{3}}\\ \\ \mathrm{sen\,}x=\dfrac{\sqrt{3}}{2}$


Como $x$ é do segundo quadrante então temos que

$\boxed{x=\dfrac{2\pi}{3}}$


$\mathrm{tg\,}\dfrac{x}{2}\\ \\ =\mathrm{tg\,}\dfrac{\,^{2\pi}\!\!\!\diagup\!\!_{3}}{2}\\ \\ =\mathrm{tg\,}\dfrac{2\pi}{6}\\ \\ =\mathrm{tg\,}\dfrac{\pi}{3}\\ \\ =\sqrt{3}\\ \\ \\ \boxed{\mathrm{tg\,}\dfrac{x}{2}=\sqrt{3}}$