Problem 68 - Algebra and Trigonometry

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\(/ z_{1}\) . $$ z_{1}=3+4 i, \quad z_{2}=2-2 i $$

Step-by-Step Solution

Answer

Polar forms: \( z_1 = 5\ (\cos(\theta_1) + i\sin(\theta_1)) \) and \( z_2 = 2\sqrt{2}\ (\cos(\theta_2) + i\sin(\theta_2)) \). Product: \( 10\sqrt{2}(\cos(\theta_{12}) + i\sin(\theta_{12})) \). Quotients: \( \frac{5}{2\sqrt{2}}(\cos(\theta_{1/2}) + i\sin(\theta_{1/2})) \) and \( \frac{1}{5}(\cos(-\theta_1) + i\sin(-\theta_1)) \).

Step 1: Write Complex Numbers in Polar Form

To convert a complex number into polar form, we first find the magnitude and angle. For \( z_1 = 3 + 4i \), the magnitude \( r_1 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \). The angle \( \theta_1 = \tan^{-1}\left(\frac{4}{3}\right) \). Thus, \( z_1 = 5(\cos(\theta_1) + i\sin(\theta_1)) \).For \( z_2 = 2 - 2i \), the magnitude \( r_2 = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \). The angle \( \theta_2 = \tan^{-1}\left(\frac{-2}{2}\right) = -\frac{\pi}{4} \). Thus, \( z_2 = 2\sqrt{2}(\cos(\theta_2) + i\sin(\theta_2)) \).

Step 2: Multiply the Complex Numbers

To find the product \( z_1 z_2 \), multiply the magnitudes and add the angles: \( |z_1 z_2| = r_1 \cdot r_2 = 5 \times 2\sqrt{2} = 10\sqrt{2} \).\( \theta_{12} = \theta_1 + \theta_2 \).Thus, \( z_1 z_2 = 10\sqrt{2}(\cos(\theta_{12}) + i\sin(\theta_{12})) \).

Step 3: Find the Quotient \(\frac{z_1}{z_2}\)

For \( \frac{z_1}{z_2} \), divide the magnitudes and subtract the angles: \( \left|\frac{z_1}{z_2}\right| = \frac{r_1}{r_2} = \frac{5}{2\sqrt{2}} \).\( \theta_{1/2} = \theta_1 - \theta_2 \).Thus, \( \frac{z_1}{z_2} = \frac{5}{2\sqrt{2}}(\cos(\theta_{1/2}) + i\sin(\theta_{1/2})) \).

Step 4: Find the Quotient \(\frac{1}{z_1}\)

For \( \frac{1}{z_1} \), take the reciprocal of the magnitude and negate the angle:\( \left|\frac{1}{z_1}\right| = \frac{1}{r_1} = \frac{1}{5} \).\( \theta_{-1} = -\theta_1 \).Thus, \( \frac{1}{z_1} = \frac{1}{5}(\cos(-\theta_1) + i\sin(-\theta_1)) \).

Key Concepts

Complex Number Multiplication

Multiplying complex numbers can be made simple with polar form. The process essentially involves multiplying their magnitudes and adding their angles.
In the given exercise, you have two complex numbers: \( z_1 = 3 + 4i \) and \( z_2 = 2 - 2i \). When you convert each to polar form, you get their magnitudes and angles, which are crucial for multiplication.

• Magnitude: For \( z_1 \), the magnitude is \( r_1 = 5 \) and for \( z_2 \), it is \( r_2 = 2\sqrt{2} \).
• Angles: \( \theta_1 \) for \( z_1 \) and \( \theta_2 = -\frac{\pi}{4} \) for \( z_2 \).

To find the product \( z_1 z_2 \), use:

• Multiply the magnitudes: \( |z_1 z_2| = 5 \times 2\sqrt{2} = 10\sqrt{2} \).
• Add the angles: \( \theta_{12} = \theta_1 + \theta_2 \).

This gives you the product in polar form: \( z_1 z_2 = 10\sqrt{2} (\cos(\theta_{12}) + i\sin(\theta_{12})) \).
This method simplifies complex multiplication to basic arithmetic of polar components.

Complex Number Division

Dividing complex numbers in polar form involves dividing magnitudes and subtracting angles. This mirrors division in the real number system but adapted for the complex plane.
Let's use the same complex numbers \( z_1 = 3 + 4i \) and \( z_2 = 2 - 2i \), now converted to polar form.

• Magnitudes: \( r_1 = 5 \) for \( z_1 \), \( r_2 = 2\sqrt{2} \) for \( z_2 \).
• Angles: \( \theta_1 \), and \( \theta_2 = -\frac{\pi}{4} \).

The division \( \frac{z_1}{z_2} \) is found by:

• Dividing the magnitudes: \( \left|\frac{z_1}{z_2}\right| = \frac{5}{2\sqrt{2}} \).
• Subtracting the angles: \( \theta_{1/2} = \theta_1 - \theta_2 \).

Thus, the quotient in polar form is: \( \frac{z_1}{z_2} = \frac{5}{2\sqrt{2}} (\cos(\theta_{1/2}) + i\sin(\theta_{1/2})) \).
Using polar form makes the division process clear and manageable.

Complex Number Reciprocals

Finding reciprocals of complex numbers in polar form is straightforward. It requires taking the reciprocal of the magnitude and negating the angle.
Consider \( z_1 = 3 + 4i \). Its polar form components are crucial:

• Magnitude: \( r_1 = 5 \).
• Angle: \( \theta_1 \).

To find the reciprocal \( \frac{1}{z_1} \):

• Magnitude: Take the reciprocal: \( \left|\frac{1}{z_1}\right| = \frac{1}{5} \).
• Angle: Negate the angle: \( \theta_{-1} = -\theta_1 \).

The result is a new polar form of the reciprocal: \( \frac{1}{z_1} = \frac{1}{5} (\cos(-\theta_1) + i\sin(-\theta_1)) \).
This polar method offers clarity and simplicity when finding the reciprocal, as it breaks the problem down to basic arithmetic.