Problem 72 - Contemporary Precalculus

Question

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$(1-i)(2 \sqrt{3}-2 i)(-4-4 \sqrt{3} i)$$

Step-by-Step Solution

Answer

Question: Find the product of the three complex numbers (1 - i), (2√3 - 2i), and (-4 - 4√3i), and express the answer in polar form. Answer: The product of the three complex numbers is 32√2cis(-π/12).

Step 1: Convert each complex number to polar form

First, let's find the polar form of each complex number: 1. \(1-i\): The magnitude is \(r=\sqrt{Re^2+Im^2}=\sqrt{1^2+(-1)^2}= \sqrt{2}\), and the argument is \(\theta = \arctan(\frac{Im}{Re})=\arctan(\frac{-1}{1})=-\frac{\pi}{4}\). Therefore, its polar form is: \(\sqrt{2}\mathrm{cis}(-\frac{\pi}{4})\). 2. \(2\sqrt{3}-2i\): The magnitude is \(r=\sqrt{Re^2+Im^2}=\sqrt{(2\sqrt{3})^2+(-2)^2}= 4\), and the argument is \(\theta = \arctan(\frac{Im}{Re})=\arctan(\frac{-2}{2\sqrt{3}})=-\frac{\pi}{6}\). Its polar form is: \(4\mathrm{cis}(-\frac{\pi}{6})\). 3. \(-4-4\sqrt{3}i\): The magnitude is \(r=\sqrt{Re^2+Im^2}=\sqrt{(-4)^2+(-4\sqrt{3})^2}= 8\), and the argument is \(\theta = \arctan(\frac{Im}{Re})+\pi=\arctan(\frac{-4\sqrt{3}}{-4})+\pi=\frac{2\pi}{3}\). Its polar form is: \(8\mathrm{cis}(\frac{2\pi}{3})\). Now that we've found the polar form of each complex number, let's move on to multiplying them.

Step 2: Multiply the polar forms

Use the polar multiplication formula: \( (r_1\mathrm{cis}(\theta_1))(r_2\mathrm{cis}(\theta_2)) = (r_1r_2)\mathrm{cis}(\theta_1+\theta_2)\). Multiply the polar forms of the given complex numbers: \( (\sqrt{2}\mathrm{cis}(-\frac{\pi}{4}))(4\mathrm{cis}(-\frac{\pi}{6}))(8\mathrm{cis}(\frac{2\pi}{3})) \) To multiply these polar forms, first multiply their magnitudes and sum their angles: Magnitude: \((\sqrt{2})(4)(8) = 32\sqrt{2}\) Angles: \(\left(-\frac{\pi}{4}\right) + \left(-\frac{\pi}{6}\right) + \frac{2\pi}{3} = -\frac{\pi}{12}\) So the product in polar form is: \(32\sqrt{2}\mathrm{cis}(-\frac{\pi}{12})\).

Key Concepts

Complex Numbers

Complex numbers are a type of number that includes both a real part and an imaginary part. They are commonly written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). This allows complex numbers to extend the real numbers into a two-dimensional space. To work with complex numbers, it's essential to understand:

• Real Part: This is the \(a\) in \(a + bi\). It is the horizontal component on the complex plane.
• Imaginary Part: This is the \(b\) in \(a + bi\). It represents the vertical component on the plane.

Representing complex numbers visually on the complex plane helps in understanding operations like addition, subtraction, and conversion into different forms. Indeed, complex numbers can be represented in rectangular form \((a, b)\) or converted into polar form for multiplication and division.

Magnitude

The magnitude of a complex number, also known as modulus or absolute value, gives us the distance of the complex number from the origin on the complex plane. To find the magnitude of a complex number \(a + bi\), we use the formula:

• \(r = \sqrt{a^2 + b^2}\)

This formula is derived from the Pythagorean theorem, as the complex number forms a right triangle on the plane with the real and imaginary parts as the legs.
In our exercise, for example, the magnitude of \(1-i\) is calculated as \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\). Knowing the magnitude is crucial, especially when converting to polar form, as it serves as the radius of the circle where the complex number lies.

Argument

The argument of a complex number is the angle it forms with the positive real axis on the complex plane. It is an essential component in the polar form representation, providing the direction of the radius measured from the origin. The argument \(\theta\) can be calculated using:

• \(\theta = \arctan\left(\frac{b}{a}\right)\)

However, this basic formula needs adjustments based on the quadrant where the complex number is located:

• For numbers in the second quadrant, add \(\pi\).
• In the third quadrant, also add \(\pi\).
• In the fourth quadrant, add \(2\pi\) or work directly with negative angles.

For example, in our exercise, the argument for \(2\sqrt{3} - 2i\) is found as \(-\frac{\pi}{6}\) using these considerations, reflecting its direction below the real axis.

Polar Multiplication

Polar multiplication is a straightforward method to multiply complex numbers represented in polar form. When a complex number is in polar form, it's expressed as \(r\mathrm{cis}(\theta)\), where \(r\) is the magnitude, and \(\theta\) is the argument. The multiplication of two complex numbers in polar form follows:

• Multiply their magnitudes: \(r_1 \cdot r_2\)
• Add their arguments: \(\theta_1 + \theta_2\)

This simplifies the process significantly compared to dealing with rectangular form directly. In the step-by-step solution, the multiplication involves computing the magnitude \(32\sqrt{2}\) and summing up the arguments: \(-\frac{\pi}{4} + -\frac{\pi}{6} + \frac{2\pi}{3} = -\frac{\pi}{12}\). The result is thus easily calculated and expressed as \(32\sqrt{2}\mathrm{cis}(-\frac{\pi}{12})\), offering clarity and precision in the arithmetic of complex numbers.