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Polar coordinates provide a unique way to express complex numbers. Instead of using the usual Cartesian coordinates (x, y), polar coordinates refer to a point by a radius and an angle. For complex numbers, a number can be represented as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle with the positive x-axis. This way of representing complex numbers can simplify multiplication and division.
When working with polar coordinates, we can easily multiply two complex numbers by multiplying their magnitudes and adding their angles. Similarly, division involves converting the magnitude by division and subtracting the angles. This is particularly useful in the given exercise, as both the numerator and the denominator are presented in this form, which allows for straightforward operations using the properties of polar coordinates.

Trigonometric functions such as cosine and sine are at the heart of converting between polar and rectangular forms. In the context of complex numbers, these functions determine the direction or the angle part of the number. For the problem at hand, knowing the trigonometric values for \( \cos \frac{5\pi}{6} \) and \( \sin \frac{5\pi}{6} \) helped in expressing the numerator, while \( \cos \frac{\pi}{6} \) and \( \sin \frac{\pi}{6} \) were needed for the denominator.
The important trigonometric values frequently used in these conversions are those of \( 30° \), \( 45° \), and \( 60° \), or their radian equivalents. For example, \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \). These allow us to transition smoothly between polar representation and operations into rectangular form, helping to simplify and calculate complex number expressions accurately.

Rectangular form of a complex number is the standard way of representing them as a sum of a real part and an imaginary part, such as \( a + bi \), where \( a \) is the real component, and \( bi \) is the imaginary component. This form is direct and visually represents the value on a two-dimensional plane.
In the exercise provided, the final aim was to express the complex division initially given in polar form, into this rectangular form. This required converting trigonometric functions into numerical values and applying operations like multiplication and division while handling the imaginary unit \( i \). Ultimately, expressing \( -12 + 12\sqrt{3}i \) in rectangular form clearly indicated the components and allowed for easy interpretation and further mathematical manipulations.