Question

Calcule a cotg 4pi/3

Answer 1

Seja $\alpha=\frac{4\pi}{3}.$ Este é um arco do terceiro quadrante, pois

$\pi <\alpha<\frac{3\pi}{2}.$


Vamos reduzir este arco ao primeiro quadrante, simplesmente subtraindo $\pi$ (meia-volta):

$\alpha-\pi=\frac{4\pi}{3}-\pi\\ \\ \alpha-\pi=\frac{4\pi-3\pi}{3}\\ \\ \alpha-\pi=\frac{\pi}{3}$


As cotangentes de $\alpha$ e de $\alpha-\pi$ são iguais:

$\mathrm{cotg\,}\alpha=\mathrm{cotg\,}(\alpha-\pi)\\ \\ \mathrm{cotg\,}\alpha=\mathrm{cotg\,}\frac{\pi}{3}\\ \\ \mathrm{cotg\,}\alpha=\dfrac{\cos \frac{\pi}{3}}{\mathrm{sen\,}\frac{\pi}{3}}\\ \\ \\ \mathrm{cotg\,}\alpha=\dfrac{(\frac{1}{2})}{(\frac{\sqrt{3}}{2})}\\ \\ \\ \mathrm{cotg\,}\alpha=\frac{1}{2}\cdot \frac{2}{\sqrt{3}}\\ \\ \mathrm{cotg\,}\alpha=\frac{1}{\sqrt{3}}\\ \\ \\ \Rightarrow \boxed{\begin{array}{c} \mathrm{cotg\,}\frac{4\pi}{3}=\frac{1}{\sqrt{3}} \end{array}}$