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We are tasked with solving the equation \( z^2 = 9e^{1.8i} \) and expressing the two solutions \( z \) in polar form, where:

- The angle \( \varphi \) lies in the interval \( -\frac{\pi}{2} \leq \varphi < \frac{3\pi}{2} \).
- \( r_1, r_2 > 0 \), indicating positive radii.

### Step-by-step process:

1. **Given equation in polar form**:
   The right-hand side is already in polar form:  
   \[
   9e^{1.8i}
   \]
   where the modulus \( r = 9 \) and the argument (angle) \( \theta = 1.8 \).

2. **General approach for solving \( z^2 = r e^{i \theta} \)**:
   We know that for any complex number \( z \) in polar form, \( z = r_0 e^{i \varphi} \). Squaring this, we get:  
   \[
   z^2 = r_0^2 e^{2i \varphi}.
   \]
   So, we need to solve for \( z \) such that \( z^2 = 9e^{1.8i} \).

3. **Find modulus and argument**:
   - The modulus of \( z^2 \) is \( 9 \), so the modulus of \( z \) (i.e., \( r_0 \)) must be \( \sqrt{9} = 3 \).
   - The argument (angle) \( \theta \) is \( 1.8 \). To find \( \varphi \), we solve:
     \[
     2\varphi = 1.8 + 2k\pi \quad \text{for integers} \ k.
     \]
     So,
     \[
     \varphi = \frac{1.8}{2} + k\pi = 0.9 + k\pi.
     \]

4. **Determine the two solutions**:
   To satisfy the condition \( -\frac{\pi}{2} \leq \varphi < \frac{3\pi}{2} \), we calculate the two possible values for \( k = 0 \) and \( k = 1 \):
   
   - For \( k = 0 \):  
     \[
     \varphi_1 = 0.9.
     \]
   
   - For \( k = 1 \):  
     \[
     \varphi_2 = 0.9 + \pi = 0.9 + 3.14 \approx 4.04.
     \]
   
   Therefore, the two angles are \( \varphi_1 = 0.9 \) and \( \varphi_2 \approx 4.04 \).

5. **Final solutions**:
   The two solutions in polar form are:
   \[
   z_1 = 3e^{0.9i}, \quad z_2 = 3e^{4.04i}.
   \]

Would you like further clarification on any step?

### Related Questions:
1. How do you convert a complex number from polar to Cartesian form?
2. Why does squaring a complex number double its argument?
3. What is the geometric interpretation of complex numbers in polar form?
4. Can a complex number have more than two square roots?
5. How does the value of \( \pi \) influence trigonometric calculations?

**Tip**: When solving complex equations, always ensure the angle is adjusted to the correct interval.

Geben Sie die beiden Lösungen z zu z^2 = 9e^{1.80i} in der Polardarstellung an.

Solve the equation z^2 = 9e^{1.80i} and express both solutions in polar form. This problem involves complex numbers, polar coordinates, and the square root of complex numbers. The detailed solution includes calculating the modulus and argument and finding the polar form solutions.

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