Problem 56 - Algebra and Trigonometry

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ z_{1}=7\left(\cos \frac{9 \pi}{8}+i \sin \frac{9 \pi}{8}\right), \quad z_{2}=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) $$

Step-by-Step Solution

Answer

Product: \(14\left(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}\right)\); Quotient: \(3.5(\cos \pi + i \sin \pi)\).

Step 1: Understand Polar Form

Polar form of a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle.

Step 2: Identify Parts of Complex Numbers

For \( z_1 = 7\left(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\right) \), we know \( r_1 = 7 \) and \( \theta_1 = \frac{9\pi}{8} \). Similarly, for \( z_2 = 2\left(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\right) \), we have \( r_2 = 2 \) and \( \theta_2 = \frac{\pi}{8} \).

Step 3: Calculate the Product \( z_1 z_2 \)

To find the product of two complex numbers in polar form, we multiply their magnitudes and add their angles. Thus, the product is:\[|z_1 z_2| = r_1\cdot r_2 = 7 \times 2 = 14\]\[\theta_{z_1 z_2} = \theta_1 + \theta_2 = \frac{9\pi}{8} + \frac{\pi}{8} = \frac{10\pi}{8} = \frac{5\pi}{4}\]So, \( z_1 z_2 = 14\left(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}\right) \).

Step 4: Calculate the Quotient \( \frac{z_1}{z_2} \)

To find the quotient of two complex numbers in polar form, we divide their magnitudes and subtract their angles. The quotient is:\[\left|\frac{z_1}{z_2}\right| = \frac{r_1}{r_2} = \frac{7}{2} = 3.5\]\[\theta_{\frac{z_1}{z_2}} = \theta_1 - \theta_2 = \frac{9\pi}{8} - \frac{\pi}{8} = \frac{8\pi}{8} = \pi\]Thus, \( \frac{z_1}{z_2} = 3.5\left(\cos \pi + i \sin \pi\right) \).

Key Concepts

Complex Numbers

Complex numbers are exciting and easy to imagine if you look at them like this: They are numbers with two parts - a real one and an imaginary one. The general representation is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part multiplied by \(i\), which is an imaginary unit with the property that \(i^2 = -1\).
These numbers can be plotted on a plane, where the x-axis is for real numbers, and the y-axis is for imaginary numbers.

• For instance, \(3 + 4i\) is a complex number where 3 is on the x-axis and 4 on the y-axis.
• The point \((3, 4)\) represents this complex number on the complex plane.

Polar form offers another perspective. Instead of focusing on horizontal and vertical distances, we measure angle and length from the origin. This way, any complex number can be expressed as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) the angle.

Multiplying Complex Numbers in Polar Form

When you multiply complex numbers in polar form, it feels like you're playing with two fun controls: one that changes size and another that changes direction. In polar form, a complex number is \(r(\cos \theta + i \sin \theta)\), and multiplication is straightforward.
To multiply two complex numbers, you:

• Multiply their magnitudes: \(|z_1 z_2| = r_1 \cdot r_2\).
• Add their angles: \(\theta_{z_1 z_2} = \theta_1 + \theta_2\).

As seen in the example, with \(z_1 = 7\left(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\right)\) and \(z_2 = 2\left(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\right)\), we get the product by calculating:

• The new magnitude: \(14\).
• The new angle: \(\frac{5\pi}{4}\).

This gives us a new complex number \(14\left(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}\right)\), a neat combination of size and direction!

Dividing Complex Numbers in Polar Form

Dividing complex numbers into polar form is not too different from multiplying them. Imagine controlling two levers again. This time, one shrinks or grows with division, and the other shifts direction with angle subtraction.
To divide two complex numbers in polar form, you:

• Divide their magnitudes: \(\left| \frac{z_1}{z_2} \right| = \frac{r_1}{r_2}\).
• Subtract their angles: \(\theta_{\frac{z_1}{z_2}} = \theta_1 - \theta_2\).

In our example, dividing \(z_1\) by \(z_2\) means:

• The magnitude becomes \(3.5\) after dividing \(7\) by \(2\).


• The new angle is \(\pi\) after subtracting \(\frac{\pi}{8}\) from \(\frac{9\pi}{8}\).

The result is \(3.5\left(\cos \pi + i \sin \pi\right)\). This is how division elegantly adjusts both size and direction! It simplifies complex divisions into easy arithmetic.