https://file.aiquickdraw.com/mathbot/file/1726673429829shwogrke.jpg

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.

Tentukan z jika arg(z + 2i) = 1/4π dan arg(z - 3) = 1/2π!

Learn how to solve for z when given two argument conditions: arg(z + 2i) = 1/4π and arg(z - 3) = 1/2π. This problem covers key concepts in complex numbers, including geometric interpretation of the argument and solving equations step-by-step. Suitable for college-level math students.

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