Question

Rewrite the rectangular form of the complex number z= - sq root 3+i in its equivalent polar form. Approximate all angle measures to the nearest degree.
z=10(cos(-30)+ i sin(-30))
z=2(cos 150+ i sin 150)
z=2(cos(-30)+ i sin(-30))
z=10 (cos 150+ i sin 150)

Answer 1

Given a complex number : $z = a + bi$

$r = \sqrt{a^2 + b^2} \\ \\ r = \sqrt{(-\sqrt{3})^2+1^2} \\ \\ r = 2$

To determine the angle:
$\theta = tan^{-1} (\frac{b}{a}) \\ \\ \theta = tan^{-1} (-\frac{1}{\sqrt{3}}) \\ \\ \theta = -30, 150$
The 'cos' term is negative and the 'sin' term is positive, therefore theta must be in 2nd quadrant.
$\theta = 150$

Final Answer:
$z = 2(cos 150 + i sin 150)$