The problem in the image asks to determine $z$ given that:

$\text{arg}(z + 2i) = \frac{1}{4}\pi \quad \text{and} \quad \text{arg}(z - 3) = \frac{1}{2}\pi.$

We will solve this step by step.

Step 1: Analyze $\text{arg}(z + 2i) = \frac{1}{4}\pi$

Let $z = x + yi$, where $x$ and $y$ are real numbers. The argument of $z + 2i$ means that we are considering $z + 2i = (x + yi) + 2i = x + (y + 2)i$. Therefore:

$\text{arg}(z + 2i) = \text{arg}(x + (y+2)i) = \frac{1}{4}\pi.$

From this, the complex number $x + (y+2)i$ forms an angle of $\frac{\pi}{4}$ with the positive real axis, which indicates that:

$\frac{y + 2}{x} = \tan\left(\frac{\pi}{4}\right) = 1.$

So we have the equation:

$y + 2 = x.$

Step 2: Analyze $\text{arg}(z - 3) = \frac{1}{2}\pi$

Next, consider $z - 3 = (x + yi) - 3 = (x - 3) + yi$. The argument $\frac{\pi}{2}$ means that the complex number $(x-3) + yi$ is purely imaginary, indicating that its real part must be zero. Thus:

$x - 3 = 0.$

This gives us:

$x = 3.$

Step 3: Solve for $y$

Now that we know $x = 3$, substitute this into the equation $y + 2 = x$ from Step 1:

$y + 2 = 3,$

which gives:

$y = 1.$

Final Answer:

Thus, the complex number $z$ is:

$z = 3 + i.$

Would you like more details on any part of the solution? Here are some related questions to deepen your understanding:

1. How is the argument of a complex number defined geometrically?
2. What does it mean when the argument of a complex number is $\frac{\pi}{2}$?
3. How would the solution change if the arguments were different angles?
4. What is the general formula for calculating the argument of a complex number?
5. Can you solve for $z$ if both arguments were given as non-standard angles (not multiples of $\pi$)?

Tip: When solving problems involving the argument of complex numbers, always visualize the geometric meaning of the angle on the complex plane.