We are tasked to determine the argument $\text{Arg}(z_1)$ given the following conditions for complex numbers $z_1$ and $z_2$:

1. $\text{Arg}\left(\frac{z_1}{z_2}\right) = \frac{\pi}{9}$
2. $\text{Arg}(z_1 z_2) = \frac{5\pi}{18}$

Step-by-Step Solution:

1. Basic properties of the argument of complex numbers:

• For the product:
  $\text{Arg}(z_1 z_2) = \text{Arg}(z_1) + \text{Arg}(z_2)$
• For the quotient:
  $\text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2)$

Let:
$\text{Arg}(z_1) = \theta_1 \quad \text{and} \quad \text{Arg}(z_2) = \theta_2.$

Thus, the given conditions become:

1. $\theta_1 - \theta_2 = \frac{\pi}{9}$
2. $\theta_1 + \theta_2 = \frac{5\pi}{18}$

2. Solve the system of equations:

We solve for $\theta_1$ (the argument of $z_1$) and $\theta_2$ (the argument of $z_2$).

Step 1: Add the two equations:
$(\theta_1 - \theta_2) + (\theta_1 + \theta_2) = \frac{\pi}{9} + \frac{5\pi}{18}$ Simplify:
$2\theta_1 = \frac{\pi}{9} + \frac{5\pi}{18}.$

Convert $\frac{\pi}{9}$ to have the same denominator as $\frac{5\pi}{18}$:
$\frac{\pi}{9} = \frac{2\pi}{18}.$

Thus:
$2\theta_1 = \frac{2\pi}{18} + \frac{5\pi}{18} = \frac{7\pi}{18}.$

Divide by 2:
$\theta_1 = \frac{7\pi}{36}.$

Final Answer:

The argument of $z_1$ is:
$\boxed{\frac{7\pi}{36}}.$

This corresponds to option (a).

Do you want me to explain any step further or clarify concepts?

Here are 5 related questions to practice:

1. How do you compute the argument for the product of two complex numbers?
2. If $\text{Arg}(z_1 z_2) = \pi$ and $\text{Arg}(z_2) = \frac{\pi}{2}$, what is $\text{Arg}(z_1)$?
3. Solve $\theta_1 - \theta_2 = \frac{\pi}{6}$ and $\theta_1 + \theta_2 = \frac{2\pi}{3}$.
4. What is the range of values for the argument of a complex number?
5. How does the argument change when a complex number is divided by another?

Tip: Always ensure angles are expressed in their principal values (from $-\pi$ to $\pi$) when solving for arguments of complex numbers.