Question

Locate the point in the complex plane and express the number in polar form.

Answer 1

Answer:

We kindly invite you to see the result in the image attached below.

The number in polar form is $z = 3\sqrt{5}\cdot e^{i\cdot 0.352\pi}$.

Step-by-step explanation:

A complex number is represented by elements of the form $a + i\,b$, for all $a$, $b$ $\in \mathbb{R}$. The first part of the sum is the real component of the complex number, whereas the second part of the sum is the imaginary component of the complex number. The real component is located on the horizontal axis and the imaginary component on the vertical axis.

Now we proceed to present the point on the graph: ($a = 6$, $b = 3$)  We kindly invite you to see the result in the image attached below.

The polar form of the complex number is defined by:

$z = r\cdot e^{i\cdot \theta}$ (1)

Where:

$r$ - Magnitude of the complex number, dimensionless.

$\theta$ - Direction of the complex number, measured in radians.

The magnitude and the direction of the complex number are defined by the following formulas:

Magnitude

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

Direction

$\theta = \tan^{-1} \frac{b}{a}$ (3)

If we know that $a = 6$ and $b = 3$, then the polar form of the number is:

$r =\sqrt{6^{2}+3^{2}}$

$r = 3\sqrt{5}$

$\theta = \tan^{-1} \frac{6}{3}$

$\theta \approx 0.352\pi \, rad$

$z = 3\sqrt{5}\cdot e^{i\cdot 0.352\pi}$

The number in polar form is $z = 3\sqrt{5}\cdot e^{i\cdot 0.352\pi}$.