5 Best Ways to Return the Angle of the Complex Argument in Radians in Python

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💡 Problem Formulation: In Python, working with complex numbers often requires finding the angle (also known as the argument) they form with the positive real axis. The task is to calculate this angle in radians for a given complex number, for instance, the complex number 3+4j would have an angle that we want to return in radians.

Method 1: Using the cmath.phase() Function

This method involves the use of the cmath module which provides access to mathematical functions for complex numbers. The phase() function specifically returns the phase of a complex number, which is the angle the number forms with the positive real axis.

Here’s an example:

import cmath

complex_num = 3 + 4j
angle = cmath.phase(complex_num)
print("The angle in radians is:", angle)

Output:

The angle in radians is: 0.9272952180016122

This code snippet imports the cmath library and then calculates the phase of the complex number 3+4j, which is observed to be approximately 0.93 radians. The cmath.phase() function is a direct way to get the angle in radians and thus is both simple and precise.

Method 2: Using the math.atan2() Function

Another approach is to use the math.atan2() function which returns the arctangent of y/x in radians. For complex numbers, y corresponds to the imaginary part and x to the real part.

Here’s an example:

import math

complex_num = 3 + 4j
angle = math.atan2(complex_num.imag, complex_num.real)
print("The angle in radians is:", angle)

Output:

The angle in radians is: 0.9272952180016122

This code utilizes the math.atan2() function to compute the angle formed by the imaginary part 4 and the real part 3 of the complex number. It’s an effective alternative to cmath.phase(), with the added flexibility of being in the math module, which is more commonly used for real number operations.

Method 3: Using Euler’s Formula

Euler’s formula states that for any real number x, e^(ix) = cos(x) + i*sin(x). By expressing a complex number in exponent form, we can infer the angle from the exponential’s exponent.

Here’s an example:

import cmath

complex_num = 3 + 4j
exponent_form = cmath.polar(complex_num)
angle = exponent_form[1]
print("The angle in radians is:", angle)

Output:

The angle in radians is: 0.9272952180016122

The code fragment uses the cmath.polar() function to convert the complex number into its magnitude and phase (angle) representation. The angle is then extracted from the tuple. This method underscores the relationship between a complex number’s polar form and its angle.

Method 4: Using NumPy’s angle() Function

When working with an array of complex numbers, NumPy’s angle() function can be particularly useful as it’s vectorized and can handle arrays efficiently, returning an array of angles for each complex number.

Here’s an example:

import numpy as np

complex_array = np.array([3 + 4j])
angles = np.angle(complex_array)
print("The angles in radians are:", angles)

Output:

The angles in radians are: [0.92729522]

This snippet demonstrates the use of NumPy’s angle() function with a NumPy array containing our complex number. This function provides a high-performance, array-oriented solution for calculating the angles in radians, which is especially efficient for computations involving large datasets.

Bonus One-Liner Method 5: Using Lambda Function

For a quick inline computation, a lambda function can be combined with the cmath.phase() function to create a one-liner that you can use in mapping or list comprehension operations.

Here’s an example:

import cmath

angle = (lambda x: cmath.phase(x))(3 + 4j)
print("The angle in radians is:", angle)

Output:

The angle in radians is: 0.9272952180016122

This code employs a lambda function that takes a complex number and immediately returns its phase using the cmath.phase() function. This serves as a compact and inline alternative which is mostly suited for quick and simple implementations.

Summary/Discussion

  • Method 1: cmath.phase() Function. Direct and specific to complex numbers. Requires importing cmath. Highly straightforward.
  • Method 2: math.atan2() Function. Accessible within the math library. Also useful for real number operations. May require additional explanation for those less familiar with trigonometry.
  • Method 3: Euler’s Formula with cmath.polar(). Conceptual approach that provides both magnitude and phase. Gives a deeper understanding of complex numbers. Slightly more verbose.
  • Method 4: NumPy’s angle() Function. Best for array operations and handling multiple complex numbers. Requires NumPy, which may be too heavy for simple operations.
  • Bonus Method 5: Lambda Function. Neat one-liner for inline calculations. Best for quick uses or when space is at a premium. Not as readable for complex operations.