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Complex numbers, often represented as \(a+bi\), can also be expressed in trigonometric form. The trigonometric form is especially helpful for multiplication and division. The key idea here is to represent the complex number in terms of its magnitude (or modulus) and argument (or angle) using trigonometric functions. Imagine a point on the complex plane with coordinates corresponding to a complex number. The distance from this point to the origin is the magnitude, and the angle it forms with the positive x-axis is the argument.

For a complex number \(z = a + bi\), its trigonometric form is \(z = r(\cos \theta + i \sin \theta)\). Here \(r\) is the magnitude, and \(\theta\) is the argument. This representation taps the essence of both geometry and trigonometry, simplifying many complex number operations by converting them into trigonometric identities.

When working with complex numbers in trigonometric form, the angle addition formulas become quite handy. They allow you to easily manipulate the trigonometric components of the complex number, especially when you multiply complex numbers together. The angle addition formulas give us a way to determine the cosine and sine of sums of angles, which is crucial in finding the form \(a + bi\).

These formulas are:

• Cosine Addition: \(\cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B\)
• Sine Addition: \(\sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B\)

In the context of the exercise, simplifying the expression involves multiplying the magnitude with the trigonometric components, but often these formulas could be applied when dealing with additional terms, just to ensure everything remains precise and clear.

Understanding the magnitude in complex numbers is essential, as it grounds the numbers in the complex plane by giving them a scale. The magnitude of a complex number \(a+bi\) is the distance from the origin \((0,0)\) to the point \((a,b)\) in the complex plane. This is calculated as \(r = \sqrt{a^2 + b^2}\).

In the trigonometric form \(z = r(\cos \theta + i \sin \theta)\), the magnitude \(r\) comes directly into play. It not only dictates the length of the line connecting the origin to the complex number but also scales the trigonometric values for cosine and sine. For example, in the problem, the initial magnitude of 2 was used to scale the components \(\cos 1.5\) and \(\sin 1.5\), grounding the complex number in the form \(a + bi\). Recognizing and properly using the magnitude facilitates a seamless translation from trigonometric form to rectangular form.