(|z| = 2)
(\arg(z) = -\frac{2}{3}\pi)
The polar form is: [ z = |z| \left( \cos(\arg(z)) + i \sin(\arg(z)) \right) ]

Given the modulus \( |z| = 2 \) and the argument \( \arg(z) = -\frac{2}{3}\pi \), we can express \( z \) in its polar form using the following formula:

\[
z = |z| \left( \cos(\arg(z)) + i \sin(\arg(z)) \right)
\]

### Substituting the given values:

\[
z = 2 \left( \cos\left(-\frac{2}{3}\pi\right) + i \sin\left(-\frac{2}{3}\pi\right) \right)
\]

### Step 1: Calculate \( \cos\left(-\frac{2}{3}\pi\right) \)
Using the cosine function:
\[
\cos\left(-\frac{2}{3}\pi\right) = \cos\left(\frac{2}{3}\pi\right) = -\frac{1}{2}
\]

### Step 2: Calculate \( \sin\left(-\frac{2}{3}\pi\right) \)
Using the sine function:
\[
\sin\left(-\frac{2}{3}\pi\right) = -\sin\left(\frac{2}{3}\pi\right) = -\frac{\sqrt{3}}{2}
\]

### Step 3: Write the complex number in polar form
Now, substituting these values into the formula:
\[
z = 2 \left( -\frac{1}{2} + i \left( -\frac{\sqrt{3}}{2} \right) \right)
\]
Simplifying:
\[
z = -1 - i\sqrt{3}
\]

Thus, the complex number in standard form is:
\[
z = -1 - i\sqrt{3}
\]

Would you like any further clarifications or details?

Here are five related questions for further exploration:
1. How do you convert from polar form to exponential form of a complex number?
2. What is the geometric interpretation of the argument of a complex number?
3. How do you multiply two complex numbers in polar form?
4. How do you find the argument of a complex number located in the third or fourth quadrant?
5. What is the relation between Cartesian form and polar form of complex numbers?

**Tip:** When working with complex numbers, always be mindful of the quadrant to correctly determine the sign of sine and cosine.

(|z| = 2), (\arg(z) = -\frac{2}{3}\pi). The polar form is: [ z = |z| \left( \cos(\arg(z)) + i \sin(\arg(z)) \right) ]

Learn how to express the complex number z with modulus |z| = 2 and argument arg(z) = -2/3π in polar form. This problem uses trigonometric functions and explores the relationship between Cartesian and polar forms of complex numbers. Suitable for high school students in Grades 10-12.

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