Question

Write a function to polar that accepts a complex number c, and returns two output arguments containing the magnitude mag and angle theta of the complex number. The output angle should be in degrees.

Short answer

To create a function named 'to_polar' that accepts a complex number \(c\), and returns the magnitude (mag) and angle (theta) in degrees, follow these steps: 1. Define the function signature: \(to\_polar(c)\), where \(c\) is a complex number. 2. Extract the real and imaginary parts of \(c\): \(a = \text{Re}(c)\) and \(b = \text{Im}(c)\). 3. Calculate the magnitude using the formula: \(\text{mag} = \sqrt{a^2 + b^2}\). 4. Calculate the angle in radians: \(\text{theta} = \arctan\left(\frac{b}{a}\right)\). 5. Convert the angle to degrees: \(\text{theta (degrees)} = \text{theta (radians)} \cdot \frac{180}{\pi}\). 6. Return the magnitude and angle in degrees as output. The function can be implemented in Python as follows: ```python import cmath import math def to_polar(c): a = c.real b = c.imag mag = math.sqrt(a**2 + b**2) theta_radians = math.atan2(b, a) theta_degrees = math.degrees(theta_radians) return mag, theta_degrees ```

Step-by-step solution

Understand complex number in rectangular form

A complex number is represented as \(c = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)). A complex number in rectangular form can be represented as a point in the complex plane with coordinates \((a, b)\).

Understand complex number in polar form

A complex number can also be represented in polar form as \(c = \text{mag} \cdot e^{i\cdot \text{theta}}\), where \(\text{mag}\) is the magnitude of the complex number and \(\text{theta}\) is the angle between the real axis and the line connecting the origin and the point representing the complex number. The magnitude can be calculated using the Pythagorean theorem: \[\text{mag} = \sqrt{a^2 + b^2}\]To find the angle in radians, we can use the arctangent function: \[\text{theta} = \arctan\left(\frac{b}{a}\right)\]To convert the angle to degrees, use the following conversion: \[\text{theta (degrees)} = \text{theta (radians)} \cdot \frac{180}{\pi}\]

Create the function to_polar

To create the function, follow these steps: 1. Define the function signature: \(to\_polar(c)\), where \(c\) is a complex number. 2. Extract the real and imaginary parts of \(c\): \(a = \text{Re}(c)\) and \(b = \text{Im}(c)\). 3. Calculate the magnitude using the formula: \(\text{mag} = \sqrt{a^2 + b^2}\). 4. Calculate the angle in radians: \(\text{theta} = \arctan\left(\frac{b}{a}\right)\). 5. Convert the angle to degrees: \(\text{theta (degrees)} = \text{theta (radians)} \cdot \frac{180}{\pi}\). 6. Return the magnitude and angle in degrees as output. Function implementation: ```python import cmath import math def to_polar(c): a = c.real b = c.imag mag = math.sqrt(a**2 + b**2) theta_radians = math.atan2(b, a) theta_degrees = math.degrees(theta_radians) return mag, theta_degrees ``` Now you have the to_polar function that will take a complex number as input and return its magnitude and angle in degrees.

Key concepts

Polar Form

When dealing with complex numbers, it's essential to know how they are represented in both rectangular and polar forms. A complex number, typically denoted as \(c = a + bi\), features two parts: the real part \(a\) and the imaginary part \(b\).
In polar form, this complex number can be rewritten as \(c = \text{mag} \cdot e^{i \cdot \text{theta}}\). The polar representation focuses on two key components:

• Magnitude (mag): The distance from the origin to the point in the complex plane.
• Angle (theta): The angle formed with the positive real axis.

Polar coordinates are incredibly useful in various applications like physics and engineering. They provide a more intuitive geometric interpretation of complex numbers, especially when dealing with rotations and oscillations.
Understanding how to convert from rectangular to polar form allows you to effectively interpret and manipulate complex numbers for practical uses.

Magnitude Calculation

To convert a complex number from rectangular to polar form, calculating the magnitude is a crucial step. The magnitude of a complex number \(c = a + bi\) is the length of the vector from the origin to the point \((a,b)\) on the complex plane.
To find this length, we apply the Pythagorean theorem:

• Formula: \(\text{mag} = \sqrt{a^2 + b^2}\)

This formula essentially calculates the hypotenuse of a right triangle where \(a\) and \(b\) are the two perpendicular sides. The magnitude tells us how far away the point representing the complex number is from the origin.
Remember, the magnitude is always a non-negative number, reflecting the absolute size of the complex number regardless of its direction in the complex plane.

Angle Calculation

With the magnitude calculated, the next step is to determine the angle \(\text{theta}\), which gives you more insight into the direction of the complex number. The angle \(\text{theta}\) is defined in relation to the real axis.
Here's how you can calculate it:

• Formula for radians: \(\text{theta} = \arctan\left(\frac{b}{a}\right)\)
• Conversion to degrees: \[\text{theta (degrees)} = \text{theta (radians)} \cdot \frac{180}{\pi}\]

This conversion is essential, as real-world applications often require the angle to be expressed in degrees rather than radians.
It's important to note that special care should be taken when \(a\) is zero, as \(\arctan\) encounters an undefined behavior. In programming, the `atan2` function is often used, which handles the quadrant information appropriately.
Overall, understanding angle calculation allows you to fully define the position of the complex number on the complex plane.