Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\left[2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]\left[2\left(\cos 225^{\circ}+i \sin 225^{\circ}\right)\right]$$

Step-by-Step Solution

Answer

The product in rectangular form is \(-4i\).

Step 1: Use the trigonometric form of complex numbers

The given expressions are in trigonometric (polar) form. We know that \( z = r(\cos \theta + i \sin \theta) \) can be written as \( z = r e^{i \theta} \).So, the given complex numbers can be expressed as:- \( z_1 = 2(\cos 45^\circ + i \sin 45^\circ) = 2e^{i(\pi/4)} \)- \( z_2 = 2(\cos 225^\circ + i \sin 225^\circ) = 2e^{i(5\pi/4)} \)

Step 2: Multiply the complex numbers

When multiplying two complex numbers in polar form, we multiply their magnitudes and add their angles: \[ z_1 \cdot z_2 = (r_1 e^{i\theta_1}) \cdot (r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)} \] Plugging in the given values: - \( r_1 = r_2 = 2 \), thus their product is \( 2 \times 2 = 4 \).- \( \theta_1 = \frac{\pi}{4} \), \( \theta_2 = \frac{5\pi}{4} \), thus their sum is \( \frac{\pi}{4} + \frac{5\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} \).- The product in trigonometric form is \( 4e^{i(3\pi/2)} \).

Step 3: Convert to rectangular form

The expression in trigonometric form \( 4e^{i(3\pi/2)} \) can be converted back to rectangular form using \( e^{i\theta} = \cos \theta + i \sin \theta \).Thus,\[ 4(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) \]- \( \cos \frac{3\pi}{2} = 0 \)- \( \sin \frac{3\pi}{2} = -1 \)Therefore, the rectangular form is:\[ 4(0 + i(-1)) = 4i(-1) = -4i \]

Step 4: Final Answer: Present the product

The product of the given complex numbers \[ \left[2 \left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right] \left[2 \left(\cos 225^{\circ}+i \sin 225^{\circ}\right)\right] \] in rectangular form is \(-4i\).

Key Concepts

Polar Form

Complex numbers can be expressed in various forms, one of which is the polar form. This form is particularly helpful when dealing with multiplications or divisions of complex numbers.
In the polar form, a complex number is represented as:

• \( z = r (\cos \theta + i \sin \theta) \)
• \( r \) is the magnitude (or modulus) of the complex number.
• \( \theta \) is the argument, typically measured in radians.

Expressing a complex number in polar form makes it easy to multiply or divide because the magnitude can be computed separately from the direction (angle). This was the starting point in our exercise.

Trigonometric Form

Closely related to the polar form, the trigonometric form is sometimes interchangeably referred to when describing complex numbers. In mathematical terms:

• \( z = r \left( \cos \theta + i \sin \theta \right) \)

This form comes into play when dealing with operations over the complex plane, like in our exercise. Here, the focus is both on the magnitude \( r \) and the angle \( \theta \). The trigonometric form makes visualizing complex numbers easier, providing a clear geometric perspective by considering both the direction and distance from the origin. These properties are useful when understanding operations like rotation and scaling in the complex plane.

Rectangular Form

The rectangular form of a complex number is perhaps the most straightforward to understand. In this form, a complex number is expressed as:

• \( z = a + bi \)
• Where \( a \) is the real part, and \( b \) is the imaginary part.

Converting between polar/trigonometric and rectangular forms involves some trigonometry. From the polar form \( z = r(\cos \theta + i \sin \theta) \), the corresponding rectangular form would be \( z = r \cos \theta + (r \sin \theta)i \). Understanding this conversion is crucial, as illustrated in our exercise where the final answer is presented in rectangular form \(-4i\), showcasing the combined impact of the magnitude and angle on the traditional x-y coordinate plane.

Multiplication of Complex Numbers

Multiplying complex numbers is notably simplified when using the polar or trigonometric forms. The process is as follows:

• Multiply the magnitudes \( r_1 \) and \( r_2 \).
• Add their angles \( \theta_1 \) and \( \theta_2 \).

This method utilizes the properties of exponents and angles and is mathematically represented as:\[ z_1 \cdot z_2 = (r_1 \cdot r_2) e^{i(\theta_1 + \theta_2)} \]This was applied in our exercise, where the multiplication transformed the original polar expressions into a new complex number in polar form. From there, it was straightforward to convert it back to rectangular form. Thus, mastering multiplication in polar form is powerful, particularly when the goal is to simplify complex number calculations efficiently.