Problem 4 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=3\left(\cos 130^{\circ}+i \sin 130^{\circ}\right) \text { and } z_{2}=4\left(\cos 170^{\circ}+i \sin 170^{\circ}\right)$$

Step-by-Step Solution

Answer

The product is approximately \(5.9978 - 10.3909i\).

Step 1: Convert Polar to Rectangular Form

First, express each complex number \( z_1 \) and \( z_2 \) in rectangular form. We use the formulas: \( z = r(\cos \theta + i \sin \theta) = r \cos \theta + r i \sin \theta \). For \( z_1 = 3(\cos 130^\circ + i \sin 130^\circ) \), calculate:\[\cos 130^\circ \approx -0.6428, \quad \sin 130^\circ \approx 0.7660\]Thus, \( z_1 = 3(-0.6428 + i \times 0.7660) = -1.9284 + 2.298i \).Similarly, for \( z_2 = 4(\cos 170^\circ + i \sin 170^\circ) \), calculate:\[\cos 170^\circ \approx -0.9848, \quad \sin 170^\circ \approx 0.1736\]Thus, \( z_2 = 4(-0.9848 + i \times 0.1736) = -3.9392 + 0.6944i \).

Step 2: Compute the Product

Next, compute the product \( z_1 z_2 \) by multiplying their rectangular forms using distributive property:\[(-1.9284 + 2.298i)(-3.9392 + 0.6944i)\]Perform the multiplication:- Real part: \((-1.9284)(-3.9392) - (2.298)(0.6944) = 7.5939 - 1.5961 = 5.9978 \)- Imaginary part: \((-1.9284)(0.6944) + (2.298)(-3.9392) = -1.338\ - 9.0529 = -10.3909 \)So, \( z_1 z_2 = 5.9978 - 10.3909i \).

Key Concepts

Polar Form

Complex numbers can beautifully express themselves in different ways. The polar form is one such representation, and it's especially useful when dealing with multiplication and division of complex numbers.
This form expresses a complex number through its magnitude and angle.
Given a complex number in polar form as \( z = r ( \cos \theta + i \sin \theta ) \), here:

• \( r \) is the magnitude (distance from the origin in the complex plane).
• \( \theta \) is the angle (in degrees or radians) with respect to the positive real axis.

For example, let's take \( z_1 = 3(\cos 130^\circ + i \sin 130^\circ) \). Here, \( r = 3 \) and \( \theta = 130^\circ \).
To understand the position, imagine the point in the complex plane which lies 3 units away from the origin and makes a 130-degree angle with the real axis.
This form is powerful, particularly for operations like multiplication, as it simplifies to just multiplying magnitudes and adding angles.

Rectangular Form

When presenting complex numbers, the rectangular form is the most common method.
This form expresses a complex number as \( z = a + bi \), where \( a \) and \( b \) are real numbers.

• \( a \) is the real part, indicating location on the horizontal (real) axis.
• \( b \) is the imaginary part, showing position on the vertical (imaginary) axis.

Converting from polar to rectangular involves calculating \( a = r \cos \theta \) and \( b = r \sin \theta \).
For instance, converting \( z_2 = 4(\cos 170^\circ + i \sin 170^\circ) \) into rectangular form gives us \( z_2 = -3.9392 + 0.6944i \).
This new form is advantageous for addition and subtraction since the components can be combined directly along the respective axes.

Multiplication of Complex Numbers

Multiplying complex numbers might seem tricky, but it becomes much easier using the right form.
In rectangular form, multiplication is straightforward by expanding using the distributive law, similar to binomials:

• Multiply the real parts: \( a \cdot c \).
• Multiply the imaginary parts: \( b \cdot d \).
• Cross-multiply real and imaginary parts, adjusting for the imaginary unit \( i^2 = -1 \).

However, it's often easier and cleaner to multiply in polar form by multiplying their magnitudes and adding their angles:
\( z_1 \cdot z_2 = (r_1 \cdot r_2)( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) ) \).
This technique significantly reduces complexity, transforming the multiplication into simple arithmetic.
In our exercise, we did the multiplication in rectangular form to arrive at the final result: \( z_1 z_2 = 5.9978 - 10.3909i \), a clear example of how different these systems can interact during calculations.