Problem 15 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=\sqrt{12}\left(\cos 350^{\circ}+i \sin 350^{\circ}\right) \text { and } z_{2}=\sqrt{3}\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)$$

Step-by-Step Solution

Answer

The quotient \( \frac{z_1}{z_2} \) in rectangular form is \( 0 - 2i \).

Step 1: Understanding Polar Form

The complex numbers are given in polar form. Polar form of a complex number is expressed as \( z = r \times (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle. Here, \( z_1 = \sqrt{12}(\cos 350^\circ + i \sin 350^\circ) \) and \( z_2 = \sqrt{3}(\cos 80^\circ + i \sin 80^\circ) \).

Step 2: Calculate the Magnitude of the Quotient

To find \( \frac{z_1}{z_2} \), divide the magnitudes. The magnitude of the quotient is \( \frac{\sqrt{12}}{\sqrt{3}} = \sqrt{4} = 2 \).

Step 3: Calculate the Angle of the Quotient

Subtract the angles of the complex numbers to find the angle for the quotient. This means \( 350^\circ - 80^\circ = 270^\circ \) (since angles wrap around at 360 degrees).

Step 4: Express the Quotient in Polar Form

Combine the magnitude and angle to express the quotient in polar form as \( 2(\cos 270^\circ + i \sin 270^\circ) \).

Step 5: Convert to Rectangular Form

Use the trigonometric values \( \cos 270^\circ = 0 \) and \( \sin 270^\circ = -1 \) to convert the expression to rectangular form. This results in \( 2(0 + i(-1)) = 0 - 2i \).

Key Concepts

Polar Form

Complex numbers can be represented in different ways. One of the most useful forms is the polar form. This is particularly handy in mathematical operations like multiplication and division. Polar form expresses a complex number in terms of a magnitude and an angle. It takes the structure:

• Magnitude, \(r\), which is the distance from the origin in the complex plane.
• Angle, \(\theta\), which is the direction measured counterclockwise from the positive x-axis.

So, a complex number in polar form is written as \(z = r \times (\cos \theta + i \sin \theta)\). This makes it easy to visualize and compute with rotation and scaling. For instance, in the original exercise, the complex number \(z_1\) is expressed as \(\sqrt{12}(\cos 350^\circ + i\sin 350^\circ)\), highlighting how the complex number is positioned relative to the origin. Recognizing a complex number in polar form is crucial for operations like division and multiplication.

Magnitude of a Complex Number

The magnitude, or modulus, of a complex number in polar form is a critical concept. It tells us how far the number lies from the origin on the complex plane. To find it, we typically employ the formula:

• \(|z| = \sqrt{x^2 + y^2}\) for a complex number \(z = x + yi\) in rectangular form.

Upon converting or working in polar coordinates, the magnitude becomes the number \(r\) in the expression \(z = r(\cos \theta + i \sin \theta)\). For example, in the given problem, the magnitudes of \(z_1\) and \(z_2\) are \(\sqrt{12}\) and \(\sqrt{3}\), respectively. When finding a quotient \(\frac{z_1}{z_2}\), we simply divide these magnitudes: \(\frac{\sqrt{12}}{\sqrt{3}} = \sqrt{4} = 2\). This simplifies the computation, transforming polar multiplication and division into simple numerical operations.

Angle Subtraction in Polar Coordinates

Angles in polar form are a key feature used for calculating the orientation of complex numbers. When dividing complex numbers in polar coordinates, the angle is determined by subtracting one angle from the other. This is akin to rotating the first angle by the negative amount of the second. This is calculated as:

• Subtract the angle of the divisor from the angle of the dividend.

In the exercise, the angles are 350° for \(z_1\) and 80° for \(z_2\). The resulting angle after subtraction is \(350^\circ - 80^\circ = 270^\circ\). In polar coordinates, subtracting angles is like adjusting the orientation of the complex number, which simplifies to operations like rotation or direction shifts in graphs. This operation is fundamental in translating polar form results back to rectangular form, where trigonometric functions such as \(\cos\) and \(\sin\) can be applied to finalize the transformation.