Problem 42 - College Trigonometry

Question

Use \(z=-\frac{3 \sqrt{3}}{2}+\frac{3}{2} i\) and \(w=3 \sqrt{2}-3 i \sqrt{2}\) to compute the quantity. Express your answers in polar form using the principal argument. $$ \frac{z}{w} $$

Step-by-Step Solution

Answer

\(\frac{z}{w} = \frac{\sqrt{3}}{4} e^{i\frac{11\pi}{12}}\) in polar form.

Step 1: Convert Complex Numbers to Polar Form

First, convert the complex numbers \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\) and \(w = 3 \sqrt{2} - 3 i \sqrt{2}\) into polar form. The polar form is \(r(\cos \theta + i \sin \theta)\) or \(re^{i\theta}\). For \(z\), calculate the modulus \(r_z = \sqrt{\left(-\frac{3 \sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2}\) and the argument \(\theta_z = \arctan \left(\frac{\frac{3}{2}}{-\frac{3 \sqrt{3}}{2}}\right)\). For \(w\), calculate the modulus \(r_w = \sqrt{(3 \sqrt{2})^2 + (-3 i \sqrt{2})^2}\) and the argument \(\theta_w = \arctan \left(\frac{-3 i \sqrt{2}}{3 \sqrt{2}}\right)\).

Step 2: Calculate Moduli and Arguments

For \(z\):- Modulus \(r_z = \sqrt{\left(-\frac{3 \sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \frac{3}{2} \sqrt{3}\).- Since \(z\) is in the second quadrant (negative real, positive imaginary), the argument \(\theta_z = \pi - \arctan \left(\frac{1}{\sqrt{3}}\right) = \frac{5\pi}{6}\).For \(w\):- Modulus \(r_w = \sqrt{(3 \sqrt{2})^2 + (-3 i \sqrt{2})^2} = 6\).- Argument \(\theta_w = -\frac{\pi}{4}\) (since \(w\) is in the fourth quadrant).

Step 3: Use Polar Forms to Divide Complex Numbers

The division of two complex numbers in polar form \(\frac{z}{w} = \frac{r_z e^{i\theta_z}}{r_w e^{i\theta_w}}\) becomes \(\frac{r_z}{r_w} e^{i(\theta_z - \theta_w)}\).Calculate:- \(\frac{r_z}{r_w} = \frac{\frac{3}{2} \sqrt{3}}{6} = \frac{\sqrt{3}}{4}\).- \(\theta_z - \theta_w = \frac{5\pi}{6} - (-\frac{\pi}{4}) = \frac{5\pi}{6} + \frac{\pi}{4} = \frac{11\pi}{12}\).Thus, \(\frac{z}{w} = \frac{\sqrt{3}}{4} e^{i\frac{11\pi}{12}}\).

Step 4: Final Expression in Polar Form

Combine the results to express \(\frac{z}{w}\) in polar form as follows:\[\frac{z}{w} = \frac{\sqrt{3}}{4} \left(\cos \frac{11\pi}{12} + i\sin \frac{11\pi}{12}\right)\].Make sure that the angle \(\frac{11\pi}{12}\) is within the principal argument range \([-\pi, \pi]\). It lies in the second quadrant, which is correct as per principal argument conventions.

Key Concepts

Polar Form

The polar form of a complex number offers a different way to express the number, relying on the angle and distance from the origin, rather than its position on the complex plane. Each complex number can be represented in the polar form as:

• Magnitude (modulus) \( r \), which is the distance from the origin to the point on the complex plane.
• Angle (argument) \( \theta \), which is measured from the positive x-axis to the line connecting the origin to the point.

This form can be expressed as \( r(\cos \theta + i \sin \theta) \), or more compactly, using Euler's formula, as \( re^{i\theta} \). The polar form is particularly useful for multiplying and dividing complex numbers, as it simplifies the calculations: - To multiply two complex numbers, you multiply their moduli and add their arguments. - To divide, you divide their moduli and subtract their arguments.In our exercise, converting complex numbers \( z \) and \( w \) to polar form is key to finding the expression for \( \frac{z}{w} \). Using polar form facilitates operations on complex numbers, allowing us to tackle even complex division with greater ease.

Modulus and Argument

The modulus and argument are fundamental components of the polar form.

Modulus

The modulus of a complex number \( z = a + bi \) is its absolute value on the complex plane, found using the formula:\[|z| = \sqrt{a^2 + b^2}\]The modulus represents the distance from the origin to the point \((a, b)\) in the complex plane.

Argument

The argument \( \theta \) of a complex number is the angle formed with the positive x-axis. It is calculated using the arctangent of the imaginary part divided by the real part: \[\theta = \arctan\left(\frac{b}{a}\right)\]However, the location of the complex number in different quadrants affects the calculation of \( \theta \). We adjust \( \theta \) depending on the sign of \( a \) and \( b \):

• If both are positive, the argument lies in the first quadrant.
• If \( a \) is negative and \( b \) is positive, it lies in the second quadrant, and we use \( \pi - \theta \).
• If both are negative, it's in the third quadrant, with \( \theta + \pi \).
• If \( a \) is positive and \( b \) is negative, it lies in the fourth quadrant, and we use \(-\theta \).

These concepts were used to derive the polar forms of \( z \) and \( w \) in the exercise.

Principal Argument

The principal argument of a complex number is the unique angle \( \theta \) that falls within the specific interval from \(-\pi\) to \(\pi\). It ensures that the representation of the angle stays within a standard range, which is crucial for consistency in computations involving complex numbers. For example, when computing the division \( \frac{z}{w} \) in the exercise, we first obtained \( \theta = \frac{11\pi}{12} \) for the final angle. This angle lies in the second quadrant and is indeed within the acceptable range for the principal argument, making it a correct representation.This principle allows us to avoid ambiguities that could arise from angles that are coterminal but not equivalent numerically. Understanding the principal argument ensures that complex numbers are consistently expressed, especially when simplifying or comparing their polar forms. Hence, verifying that the final angle for \( \frac{z}{w} \) lies within this standard range is a vital step.