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The rectangular form of a complex number is a way of expressing the number using its real and imaginary components. It is written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form is quite intuitive as it resembles plotting a point on a 2D plane, where \(a\) corresponds to the horizontal position and \(b\) corresponds to the vertical position.
One of the main advantages of the rectangular form is its simplicity during addition and subtraction. You just add or subtract the corresponding parts from each number.

• Example: To add \((3 + 4i)\) and \((1 + 2i)\), you simply add the real parts \(3 + 1\) and the imaginary parts \(4i + 2i\) to get \(4 + 6i\).
• Simplicity in visualization: Rectangular form makes it easier to visualize complex numbers as points in the complex plane.

In the exercise, we converted a complex number from polar to rectangular form, ending up with \(-\frac{3\sqrt{3}}{4} - \frac{3}{4}i\). The real part is \(-\frac{3\sqrt{3}}{4}\), and the imaginary part is \(-\frac{3}{4}\).
Understanding the rectangular form is fundamental in performing arithmetic operations on complex numbers efficiently.

The polar form of a complex number emphasizes the magnitude and angle, rather than separating the real and imaginary parts. Expressed as \(r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the magnitude or modulus, and \(\theta\) is the angle or argument of the complex number. In polar coordinates, the complex number is understood in terms of rotation and scaling.
The polar form is especially useful for multiplication and division of complex numbers, as these operations become simpler. You multiply or divide the magnitudes and add or subtract the angles.

• Convenience in Multiplication: To multiply two complex numbers, you just multiply their magnitudes and add their angles.
• Application: This form is particularly useful in fields involving waves, oscillation, and periodic functions.

In our exercise, the complex number was initially given in polar form as \(\frac{3}{2}(\cos \left(\frac{7\pi}{6}\right) + i \sin \left(\frac{7\pi}{6}\right))\). Here, the magnitude is \(\frac{3}{2}\) and the angle is \(\frac{7\pi}{6}\).
Polar representation integrates the concept of magnitude with direction, providing a different perspective in analyzing complex numbers.

The trigonometric form of a complex number is closely related to polar form but focuses on using trigonometric functions sine and cosine. It is denoted as \(r(\cos(\theta) + i\sin(\theta))\). This form is synonymous with the polar form and often used interchangeably, especially when emphasizing angles or rotations.
The trigonometric form is beneficial when it comes to transformations involving rotations. It's a natural fit for situations where the angle of a complex number is significant.

• Expression with Rotation: This form highlights how a complex number can be represented as a rotation in the complex plane.
• Clarity in Angles: Provides explicit information about the angle through the cosine and sine functions.

For the given exercise, recognizing \(\frac{3}{2}(\cos \left(\frac{7\pi}{6}\right) + i \sin \left(\frac{7\pi}{6}\right))\) in trigonometric form allows you to directly apply trigonometric identities to find the real and imaginary components, as we did when converting to rectangular form.
Understanding and recognizing the trigonometric form helps in grasping how complex numbers relate to angles and rotations in practical applications.