Problem 70 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: The result of dividing -4i by √3 + i expressed in polar form is 2(cos(-2pi/3)+i sin(-2pi/3)).

Step 1: Convert the numerator (-4i) to polar form

First, we need to find the modulus `r` and argument `theta` for -4i: r = sqrt((-4)^2) = 4 theta = arctan(b/a) = arctan(-4/0) = -90 degrees or -pi/2 radians So, the polar form of -4i is: 4(cos(-pi/2)+i sin(-pi/2))

Step 2: Convert the denominator (sqrt(3)+i) to polar form

Next, we find the modulus `r` and argument `theta` for √3 + i: r = sqrt((√3)^2 + (1)^2) = 2 theta = arctan(b/a) = arctan(1/√3) = 30 degrees or pi/6 radians So, the polar form of √3 + i is: 2(cos(pi/6)+i sin(pi/6))

Step 3: Perform division

Now, we divide the polar form of the numerator by the polar form of the denominator: result = 4(cos(-pi/2)+i sin(-pi/2)) / 2(cos(pi/6)+i sin(pi/6)) When dividing polar forms, we divide the moduli and subtract the angles: result_modulus = 4 / 2 = 2 result_angle = -pi/2 - pi/6 = -2pi/3

Step 4: Express the answer in polar form

Using the obtained modulus and angle, we can write down the answer in polar form as: 2(cos(-2pi/3)+i sin(-2pi/3))

Key Concepts

Modulus and Argument

When working with complex numbers, understanding the modulus and argument is crucial. These two components are part of the transformation from rectangular coordinates, which use real (a) and imaginary (b) parts, to polar coordinates that utilize the radius (modulus) and angle (argument).

• Modulus: Represented by \( r \), the modulus is the distance from the complex number to the origin in the complex plane. It is calculated using the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively.
• Argument: Denoted by \( \theta \), this angle represents the direction of the complex number in the complex plane, measured from the positive x-axis. The argument can be calculated using \( \theta = \arctan\left(\frac{b}{a}\right) \). It is generally expressed in radians or degrees.

By converting the complex number to polar form, you can easily perform operations like multiplication and division, which involve the modulus and argument directly.

Complex Numbers

Complex numbers are numbers composed of a real part and an imaginary part, typically described as \( a + bi \), where \( i \) is the imaginary unit with the property \( i^2 = -1 \). These numbers expand the one-dimensional number line into a complex plane, providing a two-dimensional space.

• Real Part: The real component of the complex number, denoted as \( a \), is the horizontal coordinate in the complex plane.
• Imaginary Part: The imaginary component, represented as \( bi \), is the vertical component in the complex plane.

The use of complex numbers becomes particularly essential in fields like engineering and physics, where they can represent oscillations and waves. Utilizing polar coordinates can often simplify complex number operations, especially multiplication and division, making the transformation between different forms a powerful tool.

Division of Complex Numbers

Dividing complex numbers can seem challenging in rectangular form, but it's much simpler in polar form. To divide two complex numbers, transform them into their polar forms, which separates the task into dividing moduli and subtracting arguments.

• Dividing Moduli: Simply divide the modulus of the numerator by the modulus of the denominator. For example, if the moduli are \( 4 \) and \( 2 \), the result would be \( \frac{4}{2} = 2 \).
• Subtracting Arguments: Calculate the difference between the argument of the numerator and the argument of the denominator. Using the polar forms \( \theta_1 \) and \( \theta_2 \) for the arguments, the resulting angle is \( \theta_1 - \theta_2 \).

The resulting complex number is thus expressed in polar form, combining the divided modulus and the subtracted angle. This method provides both the simplicity and elegance needed for efficiently handling complex number operations.