Let's analyze the image.

The problem involves a complex number $Z$ represented in polar form on the complex plane with coordinates provided in terms of the real and imaginary parts.

From the diagram:

1. The real part (x-axis) is $\frac{1}{2}$.
2. The imaginary part (y-axis) is $-\frac{\sqrt{3}}{2}$.

The complex number $Z$ can be represented as: $Z = \frac{1}{2} - \frac{\sqrt{3}}{2} i$

Step 1: Calculate the modulus $|Z|$

The modulus $|Z|$ is given by: $|Z| = \sqrt{\left( \frac{1}{2} \right)^2 + \left( -\frac{\sqrt{3}}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$ Thus, $|Z| = 1$, confirming that $Z$ lies on the unit circle.

Step 2: Determine the argument $\varphi$

The argument $\varphi$ is the angle that $Z$ makes with the positive x-axis. Given that the coordinates are $\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$, we are in the fourth quadrant.

For these values: $\cos(\varphi) = \frac{1}{2} \quad \text{and} \quad \sin(\varphi) = -\frac{\sqrt{3}}{2}$ This corresponds to an angle $\varphi = \frac{5\pi}{3}$ radians (since in the fourth quadrant, we subtract from $2\pi$).

Step 3: Write $Z$ in polar form

Using $\varphi = \frac{5\pi}{3}$, we can write $Z$ as: $Z = \cos\left(\frac{5\pi}{3}\right) + i \sin\left(\frac{5\pi}{3}\right)$

Answer Selection

From the given options, the correct answer is:

• e. $Z = \cos\left(\frac{5\pi}{3}\right) + i \sin\left(\frac{5\pi}{3}\right)$

Final Answer:

Option (e) is correct.

Would you like further clarification or details?

Here are some additional questions related to this topic:

1. How do you convert a complex number from Cartesian to polar form?
2. Why does the argument $\varphi$ differ based on the quadrant?
3. How can you determine the quadrant of a complex number from its Cartesian coordinates?
4. What is the geometric interpretation of the modulus and argument of a complex number?
5. How do we apply trigonometric identities to find polar coordinates?

Tip: Remember that converting between polar and Cartesian forms often involves using trigonometric functions like cosine and sine for the respective real and imaginary parts.