Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\frac{12 \text { cis } 293^{\circ}}{6 \text { cis } 23^{\circ}}$$

Step-by-Step Solution

Answer

The product in rectangular form is \(-2i\).

Step 1: Convert to Polar Form

When working with complex numbers in the form \( a \text{ cis } \theta \), remember that \( \text{cis } \theta = \cos \theta + i \sin \theta \). Here, we have \( 12 \text{ cis } 293^{\circ} \) and \( 6 \text{ cis } 23^{\circ} \). This means they represent points on the complex plane.

Step 2: Apply Polar Division Formula

According to the division formula for polar form complex numbers \( \frac{r_1 \text{ cis } \theta_1}{r_2 \text{ cis } \theta_2} = \frac{r_1}{r_2} \text{ cis } (\theta_1 - \theta_2) \). Apply this to our problem: \( \frac{12}{6} \text{ cis } (293^{\circ} - 23^{\circ}) \).

Step 3: Simplify the Ratios and Angles

First simplify the magnitude \( \frac{12}{6} = 2 \). Next, calculate the angle: \( 293^{\circ} - 23^{\circ} = 270^{\circ} \). Hence, the polar form of the result is \( 2 \text{ cis } 270^{\circ} \).

Step 4: Convert Polar to Rectangular Form

Since \( ext{cis } 270^{\circ} = \cos 270^{\circ} + i \sin 270^{\circ} \), calculate its values: \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \). Thus, \( 2 \text{ cis } 270^{\circ} = 0 + 2(-1)i = -2i \).

Key Concepts

Complex Numbers

Complex numbers offer a fascinating way to extend our understanding of numbers beyond just real numbers. They have a unique format:

• The real part, often represented as 'a'.
• The imaginary part, represented as 'bi', where 'i' is the square root of -1.

Combining these two parts, a complex number can be written as \( a + bi \). This enables us to perform calculations that aren’t possible with just real numbers. Complex numbers are especially useful in fields like engineering, physics, and mathematics, thanks to their versatility in representing quantities with two dimensions.

Rectangular Form

The rectangular form of a complex number is one of the most common ways of expressing complex numbers. This form is written as \( a + bi \), directly showing the real part and the imaginary part.

• Here, 'a' represents the position along the real (horizontal) axis.
• 'bi' represents the position along the imaginary (vertical) axis.

For example, the complex number \( -2i \) in rectangular form can be understood as having a real part of 0 and an imaginary part of -2. This means it lies purely on the imaginary axis, specifically 2 units downwards. Converting from the polar form to the rectangular form involves calculating these two components based on magnitude and angle, which makes it particularly handy for straightforward calculations and graphical representation.

Polar Form

The polar form of a complex number highlights its magnitude and angle from the positive real axis.

• Written as \( r \text{cis} \theta \), where \( r \) is the modulus or magnitude.
• The angle \( \theta \) is measured in degrees or radians from the positive real axis.

This comes particularly useful in multiplying and dividing complex numbers, as it uses the principles of trigonometry to simplify operations. Transformation between polar and rectangular forms uses the relationships:

• For amplitude \( r = \sqrt{a^2 + b^2} \)
• For angle \( \theta = \tan^{-1}(\frac{b}{a}) \)

Understanding polar form can simplify complex operations like multiplication, division, and finding powers or roots of complex numbers. For instance, in the given exercise, the product is divided using easier operations on the magnitude and angle.

Complex Plane

The complex plane provides a geometric interpretation of complex numbers. It operates similarly to a coordinate plane but specifically for complex numbers.

• The horizontal axis represents real numbers.
• The vertical axis represents imaginary numbers.

Each complex number corresponds to a unique point or vector in this plane, where position reflects both its real and imaginary parts. For example, the number \( -2i \) is located 2 units along the negative imaginary axis. Converting between rectangular and polar forms equates to moving between Cartesian coordinates and polar coordinates on this plane. This visualization helps in understanding operations like addition, subtraction, and multiplication as geometric transformations. The complex plane also simplifies visualizing complex number relationships, offering intuitive insights into their behavior.