## Quick Math Polar to Rectangular Coordinates

##### Enter expression in this form

Example (Polar form): Enter expression as e^(atan(1/2)*i) * sqrt(18)

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

In complex number theory, converting from polar coordinates to rectangular (Cartesian) coordinates is essential for simplifying many mathematical operations and visualizing complex numbers. This conversion allows us to express complex numbers in terms of their real and imaginary parts.

#### 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.

#### 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.

#### Conversion Formulas

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

**1. Real Part:**

\[ x = r \cos \theta \]

**2. Imaginary Part:**

\[ y = r \sin \theta \]

#### Worked Example

Let’s convert the complex number \( z = 5 (\cos 0.93 + i \sin 0.93) \) from polar to rectangular coordinates.

**1. Identify the Magnitude and Angle:**

– \( r = 5 \)

– \( \theta = 0.93 \) radians

**2. Calculate the Real Part:**

\[ x = r \cos \theta = 5 \cos 0.93 \]

Using a calculator:

\[ x \approx 5 \times 0.599 = 2.995 \]

**3. Calculate the Imaginary Part:**

\[ y = r \sin \theta = 5 \sin 0.93 \]

Using a calculator:

\[ y \approx 5 \times 0.802 = 4.01 \]

Therefore, the rectangular form of the complex number is:

\[ z \approx 2.995 + 4.01i \]

#### Verification

To verify, we can compare the calculated rectangular coordinates with the given polar coordinates. We started with \( r = 5 \) and \( \theta = 0.93 \) radians, and our results are \( x \approx 2.995 \) and \( y \approx 4.01 \), which align with the polar form given.

Converting complex numbers from polar to rectangular coordinates involves calculating the real and imaginary parts using straightforward trigonometric functions. This conversion is useful for visualizing complex numbers on the complex plane and simplifying mathematical operations involving complex numbers. Understanding both representations and the ability to switch between them is fundamental in complex number theory.