Question

Write a function to_complex that accepts two input arguments containing the magnitude mag and angle theta of the complex number in degrees, and returns the complex number c.

Short answer

To create the function `to_complex(mag, theta)`, first define the function with the input arguments magnitude `mag` and angle `theta`. Convert the angle from degrees to radians, using `theta_radians = (theta * math.pi) / 180`. Then, calculate the real and imaginary parts of the complex number by using the polar form formula `c = mag * (cos(theta_radians) + i * sin(theta_radians))`. Finally, return the complex number using the `complex()` function with the calculated real and imaginary parts. Here's the complete function: ```python def to_complex(mag, theta): import math theta_radians = (theta * math.pi) / 180 real_part = mag * math.cos(theta_radians) imag_part = mag * math.sin(theta_radians) return complex(real_part, imag_part) ```

Step-by-step solution

1. Define the function

First, we need to define a function called "to_complex" that takes two input arguments - the magnitude "mag" and the angle in degrees "theta". def to_complex(mag, theta): pass

2. Convert angle from degrees to radians

Inside the function, calculate the angle in radians using the given formula and store it in a variable, e.g., "theta_radians". def to_complex(mag, theta): theta_radians = (theta * math.pi) / 180

3. Calculate the real and imaginary parts

Using the polar form of complex numbers and the calculated angle in radians, find the real and imaginary parts of the complex number. def to_complex(mag, theta): theta_radians = (theta * math.pi) / 180 real_part = mag * math.cos(theta_radians) imag_part = mag * math.sin(theta_radians)

4. Return the complex number

Finally, return the complex number created using the calculated real and imaginary parts. def to_complex(mag, theta): import math theta_radians = (theta * math.pi) / 180 real_part = mag * math.cos(theta_radians) imag_part = mag * math.sin(theta_radians) return complex(real_part, imag_part) Now we have a function that takes the magnitude and angle of a complex number in degrees and returns the corresponding complex number.

Key concepts

MATLAB Functions

When working with MATLAB, understanding how functions operate is essential. Functions in MATLAB allow you to execute a sequence of statements with a specific purpose. The function encapsulates code logic, which can be reused multiple times with different inputs. In this case, the function `to_complex` was created for converting polar coordinates to a complex number. To define a function in MATLAB, simply use the keyword `function` followed by the output variable, the function name, and the input arguments in parentheses. This setup ensures that your function can handle the necessary operations and return desired results. In Python, the `def` keyword is used similarly to MATLAB's `function` keyword. It marks the start of a function definition, and enables users to build a task-specific block of code.

Polar to Rectangular Form

Converting polar to rectangular form is a frequent requirement in mathematics and engineering. Polar form describes a complex number using its magnitude and angle, while rectangular form uses Cartesian coordinates with real and imaginary parts. This conversion is accomplished by applying trigonometric functions: the real part is derived from the magnitude multiplied by the cosine of the angle, and the imaginary part from the magnitude multiplied by the sine. Having this conversion tool makes it easy to transition between different representation forms for complex numbers, which is especially useful in fields like electrical engineering and control systems.

Angle Conversion

Angles in mathematics are often dealt with in degrees or radians. In programming and mathematical computations, radians are commonly used because they simplify calculations in many trigonometric functions.The formula to convert an angle from degrees to radians is simple: multiply the angle in degrees by $\frac{\pi }{180}$. This is crucial in programs where functions like `cos` and `sin`, inherent to most programming languages, expect input in radians.Accurate angle conversion ensures that the transformation between polar and rectangular forms of complex numbers is correct. It's a fundamental operation in related mathematical and engineering tasks.

Complex Number Representation

Complex numbers have a unique representation combining both real and imaginary parts. In the complex plane, they are represented as points with coordinates. Each number is of the form $a+bi$, where $a$ is the real part and $b$ is the imaginary part.Different representation forms, such as polar (magnitude and angle) and rectangular (real and imaginary parts), are used based on the computational task at hand. Transforming complex numbers from polar to rectangular coordinates helps visualize and solve equations or computations involving these numbers more intuitively.In many programming environments, a `complex` type is provided, which directly handles these dual-nature entities, allowing for streamlined calculations and manipulations.