Problem 109 - Precalculus with Limits

Question

Given two complex numbers \(z_1 = r_1(\cos\ \theta_1 + i\ \sin\ \theta_1)\) and \(z_2 = r_2(\cos\ \theta_2 + i\ \sin\ \theta_2)\), \(z_2 \neq 0\), show that \(\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) +\ i\ \sin(\theta_1 - \theta_2)]\).

Step-by-Step Solution

Answer

The relation \(\frac{z_1}{z_2} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))\) is proven correct. This relates the division of two complex numbers with the division of their magnitudes multiplied by the cosine and sine of the difference of their arguments.

Step 1: Express \(z_1\) and \(z_2\) in terms of Euler's formula

Complex numbers in polar form can be expressed using Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\). So, represent the numbers in this form: \(z_1 = r_1e^{i\theta_1}\) and \(z_2 = r_2e^{i\theta_2}\).

Step 2: Divide the complex numbers

Using these representations of \(z_1\) and \(z_2\), calculate \(\frac{z_1}{z_2}\):\(\frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}}\).

Step 3: Simplify the calculation

Simplify the expression using the rules of division for exponential functions, which state that \(\frac{a^n}{a^m} = a^{n-m}\):\(\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}\).

Step 4: Express the result in terms of trigonometric functions

Then, convert the result back into the trigonometric form: \(\frac{z_1}{z_2} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))\).

Step 5: Interpret the result

The division of \(z_1\) by \(z_2\) equals the division of their modules multiplied by a trigonometric function of the differences of their arguments. Thus, the original statement is proved.

Key Concepts

Polar Form

The polar form of a complex number is a way to express the number using its magnitude and direction instead of its rectangular coordinates. A complex number is typically written as \( a + bi \) in rectangular form, where \( a \) is the real part and \( b \) is the imaginary part. In polar form, however, it is represented as \( re^{i\theta} \) or \( r(\cos \theta + i\sin \theta) \). Here:

• \( r \) is the magnitude or absolute value of the complex number, calculated as \( \sqrt{a^2 + b^2} \).
• \( \theta \) is the argument or angle, given by \( \tan^{-1} \left(\frac{b}{a}\right) \).

This form is particularly useful for multiplication and division because it simplifies the operations significantly by harnessing the properties of angles. For example, multiplying two numbers in polar form involves multiplying their magnitudes and adding their angles. This is why polar representation is a preferred method in many applications involving complex numbers.

Euler's Formula

Euler's formula is a fundamental equation in complex number theory that relates complex exponentials to trigonometric functions. It states that \( e^{i\theta} = \cos \theta + i\sin \theta \). This formula is crucial for expressing complex numbers in polar form.
When dealing with complex number arithmetic, Euler's formula simplifies calculations. For example, representing complex numbers using Euler's formula allows for straightforward multiplication and division by utilizing exponential rules. In calculations, operations like exponentiation translate into simpler arithmetic on the magnitude and phase angle, which would be cumbersome in standard rectangular form.
Using Euler's formula, the expression of complex numbers like \( z_1 = r_1e^{i\theta_1} \) and \( z_2 = r_2e^{i\theta_2} \) become more manageable for operations such as finding their quotient, as seen in the original exercise. It empowers one to manipulate and understand complex numbers with greater ease and flexibility.

Trigonometric Functions

Trigonometric functions play a key role when working with complex numbers in polar form. These functions, specifically sine and cosine, are fundamental in expressing the complex exponential form.
In the context of complex numbers, converting between exponential and trigonometric forms often involves using these functions. The expression \( \cos(\theta) + i\sin(\theta) \) links directly to Euler's formula, showing the equivalence to \( e^{i\theta} \). This is essential for tasks like calculating the quotient of complex numbers, as it allows transitioning back from exponential to trigonometric expressions.
Moreover, trigonometric identities help in simplifying complex number operations. For instance, when dividing two complex numbers written with trigonometric expressions of angles, these identities facilitate combining and reducing terms, making the overall computation process simpler and more efficient.