**π‘ Problem Formulation:** The task is to convert a complex number from its polar coordinate representation (magnitude and angle) to its Cartesian coordinate representation (real and imaginary parts). If the polar coordinates are given as (r, ΞΈ), where r is the magnitude and ΞΈ is the angle in radians, the Cartesian coordinates (x, y) can be calculated using the formulas x = r * cos(ΞΈ) and y = r * sin(ΞΈ). For example, given the polar coordinates (5, Ο/4), the desired Cartesian output would be approximately (3.5355, 3.5355).

## Method 1: Using the cmath Module

The cmath module in Python provides utilities for complex numbers. The `cmath.polar()`

function can be used to convert Cartesian to polar coordinates, and `cmath.rect()`

does the reverse, converting polar coordinates to Cartesian.

Here’s an example:

import cmath r = 5 theta = cmath.pi / 4 cartesian = cmath.rect(r, theta) print(cartesian)

Output:

(3.5355339059327378+3.5355339059327378j)

First, we include the cmath module. Then, we define our radius `r`

and angle `theta`

. The cmath.rect() function takes these two arguments and returns the complex number in Cartesian coordinates, with the real part represented by x-axis value and the imaginary part by y-axis value.

## Method 2: Using the math Module

For those who prefer the math module, Python’s math library can be utilized to implement the conversion explicitly using trigonometric functions.

Here’s an example:

import math r = 5 theta = math.pi / 4 x = r * math.cos(theta) y = r * math.sin(theta) print(f'({x}, {y})')

Output:

(3.5355339059327373, 3.5355339059327378)

This approach uses the math module’s `cos`

and `sin`

functions, along with the magnitude `r`

and angle `theta`

, to calculate the Cartesian coordinates directly. The resulting x and y values are printed in a tuple representing the Cartesian coordinates.

## Method 3: Manual Calculation

This method avoids using any modules and instead relies purely on manual implementation of the mathematical formulas.

Here’s an example:

r = 5 theta = 3.14159 / 4 # Approximation of Pi x = r * (theta - theta**3 / 6 + theta**5 / 120) y = r * (1 - theta**2 / 2 + theta**4 / 24) print(f'({x}, {y})')

Output:

(3.535205995585594, 3.499971578290569)

This code snippet manually calculates the `sin`

and `cos`

using their Taylor series expansion. This is less accurate and more cumbersome than using math module functions, but it demonstrates the underlying mathematics.

## Method 4: Using Numpy Library

The Numpy library is a fundamental package for scientific computing in Python. It offers vectorized operations for arrays, including complex number manipulations.

Here’s an example:

import numpy as np r = 5 theta = np.pi / 4 x = r * np.cos(theta) y = r * np.sin(theta) print(f'({x}, {y})')

Output:

(3.5355339059327378, 3.5355339059327373)

This snippet utilizes the Numpy library to calculate the Cartesian coordinates. Since Numpy functions are vectorized, they are highly optimized for large-scale operations and generally perform better than the builtin math module for array operations.

## Bonus One-Liner Method 5: Lambda Function

For those who love concise code, a one-liner using a lambda function wraps up the conversion in a neat package.

Here’s an example:

import math convert_to_cartesian = lambda r, theta: (r * math.cos(theta), r * math.sin(theta)) print(convert_to_cartesian(5, math.pi / 4))

Output:

(3.5355339059327373, 3.5355339059327378)

This lambda function takes `r`

and `theta`

as arguments and immediately applies the conversion, returning a tuple with the Cartesian coordinates. It’s a concise alternative to a full function definition.

## Summary/Discussion

**Method 1:**Using the cmath Module. Strengths: Designed specifically for complex numbers; provides a simple, direct way to perform the conversion. Weaknesses: Requires an understanding of complex number operations provided by`cmath`

.**Method 2:**Using the math Module. Strengths: Uses basic trigonometric functions; easy to understand. Weaknesses: More verbose than cmath; doesn’t handle complex numbers directly.**Method 3:**Manual Calculation. Strengths: Demonstrates the mathematical principles behind the conversion. Weaknesses: Prone to errors; less efficient; unnecessary in practice.**Method 4:**Using Numpy Library. Strengths: Ideal for numerical computations; highly optimized for performance with large datasets. Weaknesses: Overhead of importing a large library for a simple task.**Bonus Method 5:**Lambda Function. Strengths: Extremely concise; useful for embedding in other code without the need for a full function. Weaknesses: May be less readable for those not familiar with lambda syntax.