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Starting with the concept of cis notation, it's a compact and efficient way to represent complex numbers.

Let's break it down: the 'cis' stands for cosine plus i times sine, where i is the imaginary unit. This notation is particularly useful when dealing with complex numbers in polar form. For instance, when you see an expression like \( r\operatorname{cis}\theta \), it indicates that a complex number is located a distance r from the origin (which is called the modulus) and at an angle \(\theta\) (the argument) from the positive x-axis.

To convert such a number into Cartesian form, which consists of real and imaginary parts, you simply evaluate the cosine and sine of the argument and multiply them by the modulus. In our example, the modulus (r) is 9 and, after evaluating, we found that \(\cos(90^\circ) = 0\) and \(\sin(90^\circ) = 1\). This yields the Cartesian form of \(0 + 9i\).

The polar form of complex numbers is another way of expressing complex numbers, which is immensely helpful when performing multiplication or division.

The General Form

In general, the polar form of a complex number is expressed as \( r ( \cos \theta + i \sin \theta ) \) or, using 'cis' notation, \( r\operatorname{cis}\theta \). The value of r is the modulus, showing how far away the number is from the origin, and \(\theta\) is the argument, indicating the angle the line makes with the positive x-axis.

Translating into Cartesian coordinates, you create an ordered pair where the x-coordinate is \( r\cos \theta \) and the y-coordinate is \( r\sin \theta \). This corresponds to the real and imaginary parts in the Cartesian form, respectively.

Trigonometric functions are the bridge between angular measures and lengths of sides in right-angled triangles. They are essential in the context of complex numbers as they relate the polar and Cartesian forms.

The Basic Functions

Three primary trigonometric functions are most commonly used: sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). For a given angle \(\theta\), these functions relate to the ratio of sides in a right-angled triangle. Specifically, \(\cos(\theta)\) represents the adjacent side over the hypotenuse, while \(\sin(\theta)\) represents the opposite side over the hypotenuse.

For example, we calculated \(\cos(90^\circ)\) and \(\sin(90^\circ)\) when converting a complex number in 'cis' form to its Cartesian equivalent. Knowing that \(\sin(90^\circ) = 1\) and \(\cos(90^\circ) = 0\) completes the picture and allows us to easily make the transition from the angle and modulus to the real and imaginary components of the complex number.