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.