Question

Escreva na forma trigonométrica o seguinte número complexo z=-4√3-4.i
A)z=8(cos 7π/6+i.sen 7π/6)
B)z=8(cos 7π/3+i.sen 7π/3)
C)z=4(cos 7π/6+i.sen 7π/6)
D)z=4(cos 7π/3+i.sen 7π/3)
E)z=2(cos π/4+i.sen π/4)

Answer 1

$\displaystyle \sf z = a+b\cdot i \\\\ z = |z|\cdot[\ cos(\theta)+i\cdot sen(\theta) \ ] \\\\ onde : \\\\ cos(\theta) = \frac{a}{|z|} \ \ \ ;\ \ \ sen(\theta) = \frac{b}{|z|}\\\\\\ |z|=\sqrt{a^2+b^2}$

Temos :

$\displaystyle \sf z=-4\cdot \sqrt{3}-4\cdot i \\\\ |z| = \sqrt{(-4\sqrt{3})^2+(-4)^2} \\\\ |z|=\sqrt{16.3+16} = \sqrt{64} = 8 \\\\\\ \left| \begin{array}{I} \displaystyle \sf cos(\theta) = \frac{-4\sqrt{3}}{8} \to cos(\theta)=\frac{-\sqrt{3}}{2} \\\\\\ \displaystyle \sf sen(\theta)=\frac{-4}{8} \to sen(\theta)=\frac{-1}{2}\end{array} \right| \to \theta = \frac{7\pi }{6} \\\\\\ Portanto :$

$\displaystyle \sf \boxed{\sf z= 8\cdot \left[cos\left(\frac{7\pi }{6}\right) + i\cdot sen\left(\frac{7\pi}{6}\right)\right] }\checkmark$

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