Problem 12 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=8\left(\cos 80^{\circ}+i \sin 80^{\circ}\right) \text { and } z_{2}=2\left(\cos 35^{\circ}+i \sin 35^{\circ}\right)$$

Step-by-Step Solution

Answer

The quotient \(\frac{z_1}{z_2}\) in rectangular form is \(2\sqrt{2} + 2i\sqrt{2}\).

Step 1: Identify the Given Complex Numbers in Polar Form

The given complex numbers are in polar form: \(z_{1} = 8(\cos 80^{\circ} + i \sin 80^{\circ})\) and \(z_{2} = 2(\cos 35^{\circ} + i \sin 35^{\circ})\). Here, \(z_1\) has a magnitude of 8 and an angle of \(80^{\circ}\), while \(z_2\) has a magnitude of 2 and an angle of \(35^{\circ}\).

Step 2: Write the Formula for Division of Complex Numbers in Polar Form

When dividing two complex numbers in polar form, use the formula: \(\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))\), where \(r_1\) and \(r_2\) are magnitudes, and \(\theta_1\) and \(\theta_2\) are angles.

Step 3: Compute the Magnitude of the Quotient

Compute \(\frac{r_1}{r_2}\) using the given magnitudes: \(\frac{8}{2} = 4\).

Step 4: Compute the Angle of the Quotient

Calculate the angle used in the quotient: \(\theta_1 - \theta_2 = 80^{\circ} - 35^{\circ} = 45^{\circ}\).

Step 5: Express the Quotient in Polar Form

Substitute the magnitude and angle into the division formula: \(\frac{z_1}{z_2} = 4(\cos 45^{\circ} + i \sin 45^{\circ})\).

Step 6: Convert the Result to Rectangular Form

Use that \(\cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}\). Therefore, \(\frac{z_1}{z_2} = 4\left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right) = 4 \times \frac{\sqrt{2}}{2} + 4i \times \frac{\sqrt{2}}{2} = 2\sqrt{2} + 2i\sqrt{2}\).

Key Concepts

Polar Form

Complex numbers can be expressed in different forms, and one of these is the polar form. This form is particularly useful when dealing with multiplication and division of complex numbers. Polar form represents a complex number in terms of its magnitude and angle. A complex number can be written in polar form as \( r(\cos \theta + i \sin \theta) \). Here:

• \( r \) represents the magnitude or absolute value of the complex number. It is the distance from the origin to the point in the complex plane.
• \( \theta \) is the angle measured counterclockwise from the positive real axis to the line segment that represents the complex number.

This form can simplify calculations like multiplication or division, where magnitudes are multiplied or divided, and angles are added or subtracted. For example, in our exercise, we found \( z_1 \) and \( z_2 \) in polar form, which made it straightforward to calculate their quotient.

Rectangular Form

The rectangular or Cartesian form of a complex number is another way to express complex numbers. It breaks the number into its component parts along the real and imaginary axes. In rectangular form, a complex number looks like \( a + bi \), where:

• \( a \) is the real part of the complex number.
• \( bi \) is the imaginary part, where \( b \) is the coefficient of \( i \).

To convert from polar to rectangular form, you can use the relationships:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

In the given exercise, after finding the quotient \( \frac{z_1}{z_2} \) using polar form, we converted it to rectangular form to get \( 2\sqrt{2} + 2i\sqrt{2} \). This gives a clearer picture of the complex number's real and imaginary components.

Complex Division

When dividing complex numbers, using their polar form is particularly beneficial. The formula for dividing complex numbers in polar form is:\[\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))\]Here is the step-by-step breakdown:

• Magnitude Division: Divide the magnitudes \( r_1 \) and \( r_2 \). In the earlier example, we have \( \frac{8}{2} = 4 \).
• Angle Subtraction: Subtract the angle \( \theta_2 \) from \( \theta_1 \). Using the previous values, \( 80^{\circ} - 35^{\circ} = 45^{\circ} \).
• Apply Polar Formula: Plug these results back into the formula to find the quotient in polar form.

Finally, you convert this polar form result back to rectangular form if needed, as seen in the exercise solution. This makes it easier to perform operations and visualize the complex number.