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Complex numbers can be expressed in polar form, which is a particularly useful representation, especially for multiplication, division, and finding powers of complex numbers. In polar form, a complex number is expressed as \[ z = r ( \cos \theta + i \sin \theta ) \],where:

• \(r\) is the magnitude (or modulus) of the complex number. It represents the distance from the origin to the point in the complex plane.
• \(\theta\) is the argument (or angle) of the complex number. It is the angle that the line from the origin to the point makes with the positive x-axis.
• \(i\) is the imaginary unit, satisfying \(i^2 = -1\).

The polar form reveals the geometric interpretation of a complex number, making it easier to visualize complex operations like rotation and scaling.

In contrast to polar form, the rectangular form of a complex number is written as \[ z = a + bi \],where:

• \(a\) is the real part of the complex number.
• \(b\) is the imaginary part of the complex number.

This form is often more intuitive as it directly shows the real and imaginary components. Complex addition and subtraction are straightforward in this form since you can simply add or subtract the real parts and the imaginary parts separately. Transitioning between rectangular and polar forms is a crucial skill, as each form has its own advantages depending on the operation being performed. The real number plane corresponds to the x-axis, and the imaginary number plane corresponds to the y-axis, which is how these components are visually plotted.

To find the quotient of two complex numbers expressed in polar form, we utilize a straightforward technique. Given \( z_1 = r_1 ( \cos \theta_1 + i \sin \theta_1 ) \) and \( z_2 = r_2 ( \cos \theta_2 + i \sin \theta_2 ) \), the quotient \( \frac{z_1}{z_2} \) is determined by dividing the magnitudes and subtracting the angles:\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]This operation is simpler in polar form since it avoids the need for rationalizing the denominator, a common requirement when using rectangular form. Therefore, division and multiplication gain simplicity thanks to the power of exponential notation, leveraging relationships between magnitudes and angles.

Trigonometric identities are powerful and helpful tools when dealing with complex numbers, especially in polar form. For example, the angle subtraction identities:

• \( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta \)
• \( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \)

These identities allow the simplification of the expressions when performing operations like division on complex numbers in polar form. They come handy to cross-check calculations and to understand how angles combine or counterbalance each other. This makes them an invaluable part of solving problems involving complex numbers, not only for theoretical purposes but also for practical computations like the one seen in the exercise.