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The polar form of a complex number is a way to express the number using a magnitude (or radius) and an angle (or direction) instead of traditional real and imaginary parts. In polar form, a complex number is written as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the distance from the origin in the complex plane, also known as the magnitude. This form is particularly useful when dealing with multiplication and division of complex numbers.

- **Magnitude**: This is denoted by \( r \), and it represents the "length" of the complex number from the origin to its position in the complex plane.- **Angle**: Denoted by \( \theta \), this angle is measured from the positive x-axis (real axis) to the line connecting the origin and the number.
Polar form simplifies calculations involving powers and roots of complex numbers, transforming intricate algebra into more manageable trigonometry.

The rectangular form of a complex number is its representation using real and imaginary components, conventionally written as \( a + bi \). In this form, \( a \) is the real part and \( b \) is the imaginary part of the complex number. Converting from polar to rectangular form involves using the relationships:

\[ a = r \cos \theta \]\[ b = r \sin \theta \]
Thus, the polar form \( r(\cos \theta + i \sin \theta) \) can be rewritten as \( a + bi \).

This conversion is essential for operations like addition and subtraction, which are straightforward in rectangular form. It's also critical for visualization as it maps directly to the standard Cartesian coordinate system (x, y).

To understand the product of complex numbers in polar form, we leverage the simplicity that polar coordinates offer. The product of two complex numbers \( z_1 \) and \( z_2 \) in polar form is given by multiplying their magnitudes and adding their angles:

\[ z_1 z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \]
This transforms the potentially intricate multiplication of \( a + bi \) forms into straightforward arithmetic and trigonometry.

- **Multiplying Magnitudes**: Multiply the magnitudes \( r_1 \) and \( r_2 \) directly.- **Adding Angles**: Simply add the angles \( \theta_1 \) and \( \theta_2 \).
This effectively "scales and rotates" the resulting complex number in the complex plane.

The angle addition formula is pivotal when working with complex numbers in polar form. When two complex numbers are multiplied, their angles (often referred to as arguments) are added, which rotates the resulting vector by each original angle sequentially.

For the product \( z_1 z_2 \), the angle is calculated as:
\[ \theta = \theta_1 + \theta_2 \]
In the given exercise, this concept helps us find the resulting angle when two complex numbers \( z_1 \) and \( z_2 \) are multiplied, turning their individual rotations into a single combined rotation.
This makes working with rotations around the origin intuitive and is a key advantage of using polar form.

The magnitude of a complex number (also called the modulus) is its distance from the origin in the complex plane. It's denoted as \( |z| \) and is found using the Pythagorean theorem for the rectangular components:

\[ |z| = \sqrt{a^2 + b^2} \]
In polar form, it's simply \( r \).
This value is crucial when converting back and forth between rectangular and polar forms and plays a vital role in finding products, as magnitudes multiply directly.

• In our case, \( z_1 = 4 \) and \( z_2 = 3 \), thus the magnitude of their product is \( 4 \times 3 = 12 \).

Understanding magnitude aids in grasping the "size" or "absolute value" of complex numbers, which remains constant despite rotations (angle changes) of the number's position.