Problem 63 - Precalculus Mathematics for Calculus

Question

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and \(1 / z_{1}\). $$z_{1}=2 \sqrt{3}-2 i, \quad z_{2}=-1+i$$

Step-by-Step Solution

Answer

The polar forms are: \(z_{1} = 4(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6}))\), \(z_{2} = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))\). The product is \(z_{1}z_{2} = 4\sqrt{2}(\cos(\frac{\pi}{12}) + i\sin(\frac{\pi}{12}))\). The quotients are \(\frac{z_{1}}{z_{2}} = 2\sqrt{2}(\cos(-\frac{5\pi}{12}) + i\sin(-\frac{5\pi}{12}))\) and \(\frac{1}{z_{1}} = \frac{1}{4}(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))\).

Step 1: Convert \(z_{1}\) to Polar Form

The complex number \(z_{1} = 2\sqrt{3} - 2i\) needs to be written in polar form. First, find the magnitude: \(|z_{1}| = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4\). Next, find the argument: \(\theta_{1} = \tan^{-1}\left(\frac{-2}{2\sqrt{3}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}\). Thus, \(z_{1} = 4\left(\cos\left(-\frac{\pi}{6}\right) + i\sin\left(-\frac{\pi}{6}\right)\right)\).

Step 2: Convert \(z_{2}\) to Polar Form

Now convert \(z_{2} = -1 + i\) to polar form. The magnitude is \(|z_{2}| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\). The argument \(\theta_{2} = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = \frac{3\pi}{4}\). Thus, \(z_{2} = \sqrt{2} \left( \cos\left(\frac{3\pi}{4}\right) + i \sin\left(\frac{3\pi}{4}\right) \right)\).

Step 3: Find the Product \(z_{1}z_{2}\)

Multiply \(z_{1}\) and \(z_{2}\) in their polar forms. The product formula for complex numbers in polar form is: \(z_{1}z_{2} = |z_{1}||z_{2}|\left(\cos(\theta_{1}+\theta_{2}) + i\sin(\theta_{1}+\theta_{2})\right)\). Therefore, \(z_{1}z_{2} = 4\sqrt{2}\left(\cos\left(-\frac{\pi}{6} + \frac{3\pi}{4}\right) + i\sin\left(-\frac{\pi}{6} + \frac{3\pi}{4}\right)\right)\). Simplifying: \(-\frac{\pi}{6} + \frac{3\pi}{4} = \frac{\pi}{12}\), so \(z_{1}z_{2} = 4\sqrt{2}\left(\cos\left(\frac{\pi}{12}\right) + i\sin\left(\frac{\pi}{12}\right)\right)\).

Step 4: Find the Quotient \(\frac{z_{1}}{z_{2}}\)

Divide \(z_{1}\) by \(z_{2}\) using their polar forms. The quotient formula for complex numbers in polar form is: \(\frac{z_{1}}{z_{2}} = \frac{|z_{1}|}{|z_{2}|}\left(\cos(\theta_{1}-\theta_{2}) + i\sin(\theta_{1}-\theta_{2})\right)\). Therefore, \(\frac{z_{1}}{z_{2}} = \frac{4}{\sqrt{2}}\left(\cos\left(-\frac{\pi}{6} - \frac{3\pi}{4}\right) + i\sin\left(-\frac{\pi}{6} - \frac{3\pi}{4}\right)\right)\). Simplifying: \(-\frac{\pi}{6} - \frac{3\pi}{4} = -\frac{5\pi}{12}\), so \(\frac{z_{1}}{z_{2}} = 2\sqrt{2}\left(\cos\left(-\frac{5\pi}{12}\right) + i\sin\left(-\frac{5\pi}{12}\right)\right)\).

Step 5: Find the Quotient \(\frac{1}{z_{1}}\)

To find \(\frac{1}{z_{1}}\), take the reciprocal of the polar form of \(z_{1}\). This is given by \(\frac{1}{|z_{1}|}\left(\cos(-\theta_{1}) + i\sin(-\theta_{1})\right)\). Therefore, \(\frac{1}{z_{1}} = \frac{1}{4}\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right)\).

Key Concepts

Complex Multiplication

When dealing with complex numbers, multiplication in polar form simplifies the process. In polar form, a complex number is expressed as \(r(\cos\theta + i\sin\theta)\). Here, \(r\) is the magnitude, and \(\theta\) is the argument or angle measured from the positive x-axis.
This form is particularly advantageous when multiplying two complex numbers. The product formula is:

• Multiply the magnitudes: \(|z_1||z_2|\).
• Add the angles: \(\theta_1 + \theta_2\).

Therefore, given two polar form numbers, the product is a new complex number with a magnitude equal to the product of the two magnitudes, and an argument that is the sum of their arguments. This means multiplying \(z_1 = 4(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6}))\) and \(z_2 = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))\) results in a magnitude \(4\sqrt{2}\) with an argument \(\frac{\pi}{12}\).
This method highlights how polar coordinates simplify complex multiplication, streamlining what would otherwise be a more cumbersome calculation with real and imaginary parts.

Complex Division

Dividing complex numbers in polar form is equally streamlined as multiplication. The division formula in polar coordinates helps simplify this operation:

• Divide the magnitudes: \(\frac{|z_1|}{|z_2|}\).
• Subtract the angles: \(\theta_1 - \theta_2\).

This formula gives a new complex number with a magnitude that's the result of dividing the two magnitudes and an angle that's the difference of their angles.
For instance, when dividing \(z_1 = 4(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6}))\) by \(z_2 = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))\), the operation simplifies to a magnitude of \(2\sqrt{2}\) and an argument of \(-\frac{5\pi}{12}\).
This reveals how the polar form reduces division of complex numbers to basic arithmetic on their magnitudes and arguments, making this a powerful approach for complex number manipulation.

Polar Coordinates

Polar coordinates offer a geometric way of representing complex numbers, emphasizing magnitude and direction instead of the traditional x and y components. A point in polar coordinates is described by a pair \((r, \theta)\), where \(r\) is the radial distance, and \(\theta\) indicates the angle from the positive x-axis.
This representation is particularly useful when complex numbers are involved because it simplifies calculations such as multiplication, division, and exponentiation. When converting a complex number like \(2\sqrt{3} - 2i\) to polar form, first compute the magnitude \(|z|\), which is the distance from the origin, followed by the angle \(\theta\) using trigonometric relationships.
Having \(r = 4\) and \(\theta = -\frac{\pi}{6}\) for this number means it can be easily manipulated within further calculations.

• Magnitude calculation: \(|z| = \sqrt{(2\sqrt{3})^2 + (-2)^2} = 4\).
• Angle determination: \(\theta = \tan^{-1}\left(\frac{-2}{2\sqrt{3}}\right) = -\frac{\pi}{6}\).

Thus, polar coordinates offer a powerful alternative to cartesian coordinates, particularly when analyzing or calculating with complex numbers.