Question

What is the complex number z=−3+3√3i represented in polar form? z=____cis(____)

Answer 1

Answer:

6cis120°

Step-by-step explanation:

The general form of a complex number z = x+iy

If x = rcos$\theta$ and y = rsin$\theta$

where r is the modulus of the complex number and $\theta$ is the argument, z in polar form is represented as:

z =  rcos$\theta$+ rsin$\theta$

z = r(cos$\theta$+ isin$\theta$)

z = rcis$\theta$

Given the complex number z=−3+3√3i

r = $\sqrt{x^{2}+y^{2} }$

$r = \sqrt{(-3)^{2}+(3\sqrt{3})^{2} } \\r = \sqrt{9+27} \\r = \sqrt{36}\\ r = 6$

$argument\\\theta = arctan \frac{y}{x} \\\theta = arctan \frac{3\sqrt{3} }{-3} \\\theta = arctan {-\sqrt{3} } \\\theta = -60 \deg$

since the argument is negative and tan is negative in the 2nd and 4th quadrant,

In the second quadrant, theta = 180- 60 = 120°

On substituting r and theta, the complex number in polar form is expressed as 6cis120°