If `|z_(1)| = sqrt(2), |z_(2)| = sqrt(3) and |z_(1) + z_(2)| = sqrt((5-2sqrt(3)))` then arg `((z_(1))/(z_(2)))` (not neccessarily principal)

Options

A. `(3pi)/(4)`
B. `(2pi)/(3)`
C. `(5pi)/(4)`
D. `(5)/(2)`

Explanation

To solve the problem, we need to find the argument of the complex number \(\frac{z_1}{z_2}\) given the magnitudes of \(z_1\) and \(z_2\), and the magnitude of their sum.

### Step-by-step Solution:

1. **Given Information:**
\[ |z_1| = \sqrt{2}, \quad |z_2| = \sqrt{3}, \quad |z_1 + z_2| = \sqrt{5 - 2\sqrt{3}} \]

2. **Using the Formula for the Magnitude of the Sum of Two Complex Numbers:**
We know that:
\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2|z_1||z_2|\cos(\theta_1 - \theta_2) \]
where \(\theta_1 = \arg(z_1)\) and \(\theta_2 = \arg(z_2)\).

3. **Calculate the Magnitudes:**
\[ |z_1|^2 = (\sqrt{2})^2 = 2, \quad |z_2|^2 = (\sqrt{3})^2 = 3 \]
Therefore,
\[ |z_1|^2 + |z_2|^2 = 2 + 3 = 5 \]

4. **Calculate \(|z_1 + z_2|^2\):**
\[ |z_1 + z_2|^2 = \left(\sqrt{5 - 2\sqrt{3}}\right)^2 = 5 - 2\sqrt{3} \]

5. **Substituting into the Magnitude Equation:**
\[ 5 - 2\sqrt{3} = 5 + 2\sqrt{6}\cos(\theta_1 - \theta_2) \]

6. **Rearranging the Equation:**
\[ -2\sqrt{3} = 2\sqrt{6}\cos(\theta_1 - \theta_2) \]
Dividing both sides by 2:
\[ -\sqrt{3} = \sqrt{6}\cos(\theta_1 - \theta_2) \]

7. **Solving for \(\cos(\theta_1 - \theta_2)\):**
\[ \cos(\theta_1 - \theta_2) = -\frac{\sqrt{3}}{\sqrt{6}} = -\frac{\sqrt{3}}{\sqrt{2}\sqrt{3}} = -\frac{1}{\sqrt{2}} \]

8. **Finding the Angles:**
The angles where \(\cos\) is \(-\frac{1}{\sqrt{2}}\) are:
\[ \theta_1 - \theta_2 = \frac{3\pi}{4} \quad \text{or} \quad \theta_1 - \theta_2 = \frac{5\pi}{4} \]

9. **Conclusion:**
Therefore, the argument of \(\frac{z_1}{z_2}\) can be expressed as:
\[ \arg\left(\frac{z_1}{z_2}\right) = \theta_1 - \theta_2 = \frac{3\pi}{4} \quad \text{or} \quad \frac{5\pi}{4} \]