Problem 54 - Precalculus Mathematics for Calculus

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\). Express your answer in polar form. $$z_{1}=\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}, \quad z_{2}=\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

Product: \(e^{i\pi}\); Quotient: \(e^{-i\frac{\pi}{2}}\).

Step 1: Understanding Complex Numbers in Polar Form

Given two complex numbers, \(z_1\) and \(z_2\), which are expressed in polar form as \(z_1 = \cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\) and \(z_2 = \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\). These can also be expressed in exponential form as \(z_1 = e^{i\theta_1}\) and \(z_2 = e^{i\theta_2}\), where \(\theta_1 = \frac{\pi}{4}\) and \(\theta_2 = \frac{3\pi}{4}\).

Step 2: Multiply the Complex Numbers

To find the product \(z_1z_2\) in polar form, use the rule for multiplying complex numbers in polar form: \(|z_1z_2| = |z_1| |z_2|\) and \(\text{arg}(z_1z_2) = \text{arg}(z_1) + \text{arg}(z_2)\). Since both \(z_1\) and \(z_2\) have magnitude 1, the magnitude of \(z_1z_2\) is 1. The argument is \(\frac{\pi}{4} + \frac{3\pi}{4} = \pi\). Thus, \(z_1z_2 = e^{i\pi}\).

Step 3: Divide the Complex Numbers

To find the quotient \(z_1/z_2\) in polar form, use the rule for dividing complex numbers: \(|z_1/z_2| = |z_1|/|z_2|\) and \(\text{arg}(z_1/z_2) = \text{arg}(z_1) - \text{arg}(z_2)\). As before, both magnitudes are 1, so the magnitude of \(z_1/z_2\) is 1. The argument is \(\frac{\pi}{4} - \frac{3\pi}{4} = -\frac{\pi}{2}\). Thus, \(z_1/z_2 = e^{-i\frac{\pi}{2}}\).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are expressed in the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). This unique property allows complex numbers to be used in a variety of mathematical and engineering applications.

However, complex numbers can also be represented in polar form, which is especially useful for multiplication and division. The polar form of a complex number is \(r(\cos \theta + i \sin \theta)\) or more concisely \(re^{i\theta}\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the argument (the angle in radians with the positive real axis). Polar form is advantageous because it simplifies the process of multiplication and division of complex numbers.

Multiplying Complex Numbers

Multiplying complex numbers in polar form is straightforward and uses basic properties of exponents. If you have two complex numbers \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\), their product is given by the rule:

• Magnitude of product: \(|z_1 z_2| = r_1 r_2\)
• Argument of product: \(\text{arg}(z_1 z_2) = \theta_1 + \theta_2\)

The final result is \(z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}\).

In the original exercise, both \(z_1\) and \(z_2\) have magnitude 1, which simplifies calculations. Thus, the product of the magnitudes is \(1 \cdot 1 = 1\), and the angles are added as follows: \(\frac{\pi}{4} + \frac{3\pi}{4} = \pi\). Therefore, the product \(z_1 z_2\) becomes \(e^{i\pi}\), which corresponds to \(-1\) in rectangular form, due to Euler's formula.

Dividing Complex Numbers

Dividing complex numbers in polar form also leverages the power of their exponential representation. For two complex numbers \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\), their quotient \(z_1 / z_2\) is calculated using:

• Magnitude of quotient: \(|z_1 / z_2| = r_1 / r_2\)
• Argument of quotient: \(\text{arg}(z_1 / z_2) = \theta_1 - \theta_2\)

Thus, \(z_1 / z_2 = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}\).

In the problem provided, the magnitudes of \(z_1\) and \(z_2\) are both 1. This ensures that the division of magnitudes is simply 1/1 = 1. The argument calculation is \(\frac{\pi}{4} - \frac{3\pi}{4} = -\frac{\pi}{2}\). So, the division in polar form results in \(e^{-i\frac{\pi}{2}}\), which translates to \(-i\) in the standard form.