Let \(z\) be complex number with modulus 2 and argument \([(-2 \pi) / 3]\) then \(\mathrm{z}=\) (a) \(-1+\mathrm{i} \sqrt{3}\) (b) \([(-1+i \sqrt{3}) / 2]\) (c) \(-1-\mathrm{i} \sqrt{3}\) (d) \([(-1-i \sqrt{3}) / 2]\)

Understanding the polar form of complex numbers adds a powerful tool to your mathematical toolkit. It is a different way to express complex numbers, which typically are written in the form of a + bi, where a is the real part, and bi is the imaginary part. In polar form, a complex number is instead defined by two parameters: its modulus (or magnitude) and its argument (or angle).

The modulus, denoted as r, represents the distance from the origin to the point in the complex plane, and the argument, denoted as θ, represents the counterclockwise angle from the positive real axis to the point. In the polar form, the complex number z is represented as r(cosθ + isinθ).

Converting to Polar Form

Given a complex number with a modulus of r and an argument of θ, you use the formula:
\[ z = r(\text{cos}θ + i\text{sin}θ) \]
This form makes certain complex number operations, such as multiplication and division, more intuitive since you can simply add or subtract the arguments and multiply or divide the moduli.

Trigonometric functions are fundamental in the study of periodic phenomena, and they play a significant role when dealing with complex numbers. The most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan), all of which can be defined using the unit circle.

The sine function represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, while the cosine represents the ratio of the adjacent side to the hypotenuse. In the context of the unit circle, for any angle θ, sinθ is the y-coordinate and cosθ is the x-coordinate of the point where the terminal side of the angle intersects the circle.

Understanding Sign Changes

It's important to note that the signs of these functions change depending on the quadrant of the angle. Moreover, they both become periodic functions with a period of 2π, meaning they repeat their values in a regular interval. Recognizing these properties allows for the simplification of complex tasks, such as determining the sine and cosine of negative angles, as seen in our exercise.

The unit circle is a circle with a radius of one unit centered at the origin of the coordinate system. It's a crucial concept for understanding trigonometric functions because it provides a geometric representation of these functions.

Each point on the unit circle corresponds to an angle and can be described using the coordinates (cosθ, sinθ). Because the length of the radius is 1, this also means that cosθ and sinθ are directly the coordinates of the point on the circle for a given angle θ.

Using the Unit Circle

By moving around the unit circle, one can easily visualize how the values of sine and cosine change with the angle. This understanding is crucial when dealing with complex numbers and especially when converting between rectangular and polar forms. In our initial example, utilizing the unit circle knowledge allowed for a quick identification of the cosine and sine of a negative angle, eventually leading to determining the correct form of the complex number z.