Problem 39 - Algebra and Trigonometry

Question

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=3 e^{i \frac{13 \pi}{18}}\) \(w=4 e^{i \frac{3 \pi}{2}}\)

Step-by-Step Solution

Answer

The product \(zw\) is \(12 e^{i \frac{20\pi}{9}}\) (polar: \( 12 \angle \frac{20\pi}{9} \)).The quotient \( \frac{z}{w} \) is \(\frac{3}{4} e^{i \frac{-7\pi}{9}}\) (polar: \(\frac{3}{4} \angle \frac{-7\pi}{9} \)).

Step 1 - Write down the given complex numbers

The given complex numbers in exponential form are:\( z = 3 e^{i \frac{13\pi}{18}} \) \( w = 4 e^{i \frac{3\pi}{2}} \)

Step 2 - Find the product of \(z\) and \(w\)

The product of two complex numbers in exponential form is found by multiplying their magnitudes and adding their angles. Therefore,\( z w = 3 e^{i \frac{13\pi}{18}} \cdot 4 e^{i \frac{3\pi}{2}} \)This simplifies to:\( z w = (3 \cdot 4) e^{i (\frac{13\pi}{18} + \frac{3\pi}{2})} = 12 e^{i (\frac{13\pi}{18} + \frac{27\pi}{18})} = 12 e^{i \frac{40\pi}{18}} = 12 e^{i \frac{20\pi}{9}} \)

Step 3 - Write \(zw\) in polar form

The polar form of \( zw \) is:\( 12 \angle \frac{20\pi}{9} \)

Step 4 - Find the quotient of \(z\) and \(w\)

The quotient of two complex numbers in exponential form is found by dividing their magnitudes and subtracting their angles. Therefore,\( \frac{z}{w} = \frac{3 e^{i \frac{13\pi}{18}}}{4 e^{i \frac{3\pi}{2}}} \)This simplifies to:\( \frac{z}{w} = \frac{3}{4} e^{i (\frac{13\pi}{18} - \frac{3\pi}{2})} = \frac{3}{4} e^{i (\frac{13\pi}{18} - \frac{27\pi}{18})} = \frac{3}{4} e^{i \frac{-14\pi}{18}} = \frac{3}{4} e^{i \frac{-7\pi}{9}} \)

Step 5 - Write \(\frac{z}{w}\) in polar form

The polar form of \( \frac{z}{w} \) is:\( \frac{3}{4} \angle \frac{-7\pi}{9} \)

Key Concepts

Exponential Form

Exponential form is a way to represent complex numbers. Instead of using real and imaginary parts, we use the magnitude and angle. The general form is given by:
ewline z = re^{i\theta}
ewline where **r** is the magnitude (or absolute value) of the complex number and **\(\theta\)** is the angle (or argument) with the positive x-axis.
To convert a complex number from rectangular form (a + bi) to exponential form, you use the following steps:

• Find the magnitude **r**: \( r = \sqrt{a^2 + b^2} \)
• Determine the angle **\(\theta\)**: \( \theta = \tan^{-1}(\frac{b}{a}) \)

For example, for the number z = 3e^{i\frac{13\pi}{18}}, 3 is the magnitude, and \(\frac{13\pi}{18}\) is the angle. This form is particularly useful in multiplying or dividing complex numbers.

Polar Form

Polar form is another way to represent complex numbers, closely related to exponential form. It uses the magnitude and angle, but it is written differently:
ewline z = r \( \angle \theta \)
ewline Similar to the exponential form, **r** is the magnitude and **\(\theta\)** is the angle. To convert between exponential form and polar form:

• From exponential to polar: \( re^{i\theta} = r \angle \theta \)
• From polar to exponential: \( r \angle \theta = re^{i\theta} \)

For example, the polar form of zw = 12e^{i\frac{20\pi}{9}} is 12 \( \angle \frac{20\pi}{9} \). Both representations are equivalent and can be used interchangeably.

Magnitude and Angle Multiplication

Multiplying complex numbers in exponential form is straightforward. You simply multiply their magnitudes and add their angles:
ewline If z = re^{i\theta} and w = se^{i\phi}, then
ewline zw = (r * s)e^{i(\theta + \phi)}.
For example, given z = 3e^{i\frac{13\pi}{18}} and w = 4e^{i\frac{3\pi}{2}}, their product in exponential form is:
ewline zw = (3 * 4)e^{i(\frac{13\pi}{18} + \frac{3\pi}{2})} = 12e^{i\frac{20\pi}{9}}.
The corresponding polar form is 12 \( \angle \frac{20\pi}{9} \). Remember to always calculate the new magnitude by multiplying the original magnitudes and find the new angle by adding the original angles.

Magnitude and Angle Division

Dividing complex numbers in exponential form is just as simple as multiplication. You divide their magnitudes and subtract their angles:
ewline If z = re^{i\theta} and w = se^{i\phi}, then
ewline \( \frac{z}{w} = \frac{r}{s}e^{i(\theta - \phi)} \).
For instance, using the given numbers z = 3e^{i\frac{13\pi}{18}} and w = 4e^{i\frac{3\pi}{2}}, their quotient in exponential form is:
ewline \( \frac{z}{w} = \frac{3}{4}e^{i(\frac{13\pi}{18} - \frac{3\pi}{2})} = \frac{3}{4}e^{i\frac{-7\pi}{9}} \).
The corresponding polar form is \( \frac{3}{4} \angle \frac{-7\pi}{9} \). When performing these operations, ensure you correctly apply the rules for angle subtraction and magnitude division.