Question

Fill in the missing values in the table. Complete first row.

Answer 1

Step-by-step explanation:

The polar form of a complex number z = x + iy is calculated using the formula below:

`z=rcis\text{ }\theta`

where:

r = sqrt root of (x² + y²) and is also considered as the modulus of the complex number

θ = tan^-1 (y/x) is the argument of the complex number.

So, for the first complex number 3 - √3i, let's convert this to polar form in which x = 3 and y = - √3.

To convert, let's solve for the "r" or the modulus first.

`\begin{gathered} r=√(x^2+y^2) \\ r=\sqrt{3^2+(-√(3))^2} \\ r=√(9+3) \\ r=√(12) \\ r=2√(3) \end{gathered}`

The modulus is 2√3.

Let's solve for the θ or the argument.

`\begin{gathered} \theta=tan^(-1)((y)/(x)) \\ \theta=tan^(-1)((-√(3))/(3)) \\ \theta=-(1)/(6)\pi \end{gathered}`

The argument is -1/6π.

Plugging in the modulus and argument to the abbreviated polar form of a complex number, the result is:

`rcis\theta\Rightarrow2√(3)cis-(1)/(6)\pi`