https://file.aiquickdraw.com/mathbot/file/1730574188617z9i6n355.jpg

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?

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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.

Based on the image, determine the complex number Z represented in polar form on the complex plane.

Learn how to solve for the complex number Z in polar form based on a visual representation in the complex plane. This problem involves calculating the modulus and argument, as well as using trigonometric identities to express Z in polar coordinates. Suitable for Grades 10-12.

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