We are tasked with finding a complex number $z$ that satisfies the following two conditions:

1. $|z - (2 + 2i)| = 2$
2. $\arg(z - (2 + 2i)) = \frac{\pi}{6}$

Step 1: Representing $z$ in terms of a shifted complex number

Let: $w = z - (2 + 2i)$ This means we are looking for $w$ such that:

• $|w| = 2$
• $\arg(w) = \frac{\pi}{6}$

Step 2: Using the polar form of complex numbers

From polar coordinates, we know that any complex number $w$ can be expressed as: $w = r(\cos \theta + i \sin \theta)$ where $r$ is the magnitude and $\theta$ is the argument of $w$.

Given that $|w| = 2$ and $\arg(w) = \frac{\pi}{6}$, we have: $w = 2\left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right)$

Step 3: Simplifying the expression for $w$

We now calculate the trigonometric values: $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin\frac{\pi}{6} = \frac{1}{2}$ Thus, $w$ becomes: $w = 2\left( \frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \right)$ $w = \sqrt{3} + i$

Step 4: Finding $z$

Since $w = z - (2 + 2i)$, we have: $z - (2 + 2i) = \sqrt{3} + i$ Solving for $z$: $z = (\sqrt{3} + i) + (2 + 2i)$ $z = \sqrt{3} + 2 + (1 + 2)i$ $z = (\sqrt{3} + 2) + (3i)$

Thus, the complex number $z$ is: $z = \sqrt{3} + 2 + 3i$

Final Answer:

$z = \sqrt{3} + 2 + 3i$

Would you like further details or have any questions?

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Here are 5 related questions:

1. How do we find the modulus of a complex number?
2. What is the geometric interpretation of the argument of a complex number?
3. How can we express a complex number in polar form?
4. How does adding a complex number shift it in the complex plane?
5. What is the connection between Euler's formula and polar representation?

Tip: When solving complex number problems, converting between rectangular and polar forms can simplify operations like multiplication, division, and finding powers or roots.