Problem 45 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\left[2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right]\left[6\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)\right]$$

Step-by-Step Solution

Answer

The product of the two complex numbers is \(12(\cos(π/3) +i \sin(π/3))\).

Step 1: Separate the magnitudes and angles

Separate the magnitudes (r-values) and the angles (θ-values) of the complex numbers. The first complex number has magnitude 2 and angle \(π/4\), and the second complex number has magnitude 6 and angle \(π/12\).

Step 2: Multiply the magnitudes and add the angles

By the rule for multiplying complex numbers in trigonometric form, we multiply the magnitudes and add up the angles. This gives us a magnitude of \(2*6=12\), and an angle of \(π/4 + π/12 = π/3\).

Step 3: Formulate the resulting complex number

Place the resulting magnitude and angle back into trigonometric form, giving us the complex number \(12(\cos(π/3) +i \sin(π/3))\).

Key Concepts

Trigonometric Form

The trigonometric form of a complex number provides a way to express complex numbers using polar coordinates.Trigonometric form is particularly useful for multiplying or dividing complex numbers.It represents a complex number as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle).
Unlike the standard form, which is \( a + bi \), this form takes advantage of the angle to simplify calculations, especially when dealing with powers or roots.Since \( \cos \) and \( \sin \) are periodic functions, they repeat values over certain intervals.

• The magnitude \( r \) reflects how far the point lies from the origin.
• The angle \( \theta \) is measured from the positive x-axis counterclockwise to the line representing the complex number.

Understanding trigonometric form is crucial for working efficiently with complex number operations.

Multiplying Complex Numbers

Multiplying complex numbers in trigonometric form can be done easily by multiplying their magnitudes and adding their angles. This builds on the polar interpretation of complex numbers.Given two complex numbers: \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), the product is found through the following steps:

• Multiply the magnitudes: The new magnitude is \( r_1 \times r_2 \).
• Add the angles: The new angle is \( \theta_1 + \theta_2 \).

This is captured in the simple formula: \[ r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\right)\]This rule harnesses the power of trigonometric identities like the angle sum formulas for sine and cosine.This not only simplifies the process but also highlights the beauty and efficiency of using trigonometric form.

Dividing Complex Numbers

Dividing complex numbers in trigonometric form involves dividing their magnitudes and subtracting their angles.This operation feels very intuitive when you understand the geometric representation.For two complex numbers of forms \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), the quotient can be calculated as:

• Divide the magnitudes: The new magnitude is \( \frac{r_1}{r_2} \).
• Subtract the angles: The new angle is \( \theta_1 - \theta_2 \).

Thus, the resulting complex number takes the form:\[ \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right)\]This subtraction of angles is due to the inverse nature of division, as opposed to addition in multiplication, and it mirrors the angle subtraction identities from trigonometry.

Polar Coordinates

Polar coordinates provide a powerful way to express complex numbers.In this system, each complex number is described by a distance and an angle rather than rectangular coordinates (real and imaginary components).The polar form is particularly helpful in simplifying the multiplication and division of complex numbers.

• Magnitude \( r \): This is equivalent to the radius in polar coordinates and tells you how far the point is from the origin.
• Angle \( \theta \): This is measured in radians and indicates the direction of the line from the origin to the point in the complex plane.

The transition between rectangular and polar forms enables diverse analyses and operations on complex numbers.For instance, using polar coordinates, we can smoothly transition not only into multiplication and division but also into understanding complex dynamics like roots and powers more deeply.