## Quick Math Rectangular to Polar Coordinates

##### Enter expression in this form

Example (rectangular form): Enter expression as 3 + 4i

### Conversion of Rectangular to Polar Coordinates in Complex Numbers

In complex number theory, converting from rectangular (Cartesian) coordinates to polar coordinates is a common task. This conversion allows us to represent complex numbers in terms of their magnitude (or modulus) and angle (or argument).

#### Rectangular (Cartesian) Form

A complex number in rectangular form is expressed as:

\[ z = x + yi \]

where \( x \) is the real part and \( y \) is the imaginary part of the complex number.

#### Polar Form

A complex number in polar form is expressed as:

\[ z = r(\cos \theta + i \sin \theta) \]

where:

– \( r \) is the magnitude (or modulus) of the complex number.

– \( \theta \) is the argument (or angle) of the complex number, usually measured in radians.

#### Conversion Formulas

To convert a complex number from rectangular to polar coordinates, we use the following formulas:

##### 1. Magnitude (Modulus):

\[ r = \sqrt{x^2 + y^2} \]

##### 2. Argument (Angle):

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Note: The angle \( \theta \) should be adjusted according to the quadrant in which the complex number lies.

#### Worked **Example**

Let’s convert the complex number \( z = 3 + 4i \) from rectangular to polar coordinates.

**1. Calculate the Magnitude:**

\[

r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

\]

**2. Calculate the Argument:**

\[

\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{4}{3}\right)

\]

**Using a calculator to find the angle:**

\[

\theta \approx \tan^{-1}(1.3333) \approx 0.93 \text{ radians}

\]

Thus, the polar form of the complex number \( z = 3 + 4i \) is:

\[

z = 5 (\cos 0.93 + i \sin 0.93)

\]

#### Verification

To verify, we can use the polar form to find the rectangular coordinates again and ensure consistency.

From the polar form \( z = 5 (\cos 0.93 + i \sin 0.93) \):

– Calculate the real part:

\[

x = 5 \cos 0.93 \approx 5 \cdot 0.599 = 3

\]

– Calculate the imaginary part:

\[

y = 5 \sin 0.93 \approx 5 \cdot 0.801 = 4

\]

So, the rectangular coordinates are \( 3 + 4i \), confirming the accuracy of our conversion.

Converting complex numbers from rectangular to polar coordinates involves calculating the magnitude and angle using straightforward formulas. This conversion is useful in many applications, such as simplifying multiplication and division of complex numbers, and understanding their geometric interpretation.