Find the set of points formed by the expression 𝜋<|𝑧−4+2𝑖|<3𝜋.

To find the set of points formed by the expression 𝜋<|𝑧−4+2𝑖|<3𝜋, we need to solve the inequality.

Let 𝑧 = 𝑥 + 𝑦𝑖, where 𝑥 and 𝑦 are real numbers.

The inequality can be written as:
𝜋 < |(𝑥 + 𝑦𝑖) − (4 + 2𝑖)| < 3𝜋

Simplifying the expression inside the absolute value:
𝜋 < |𝑥 − 4 + (𝑦 − 2)𝑖| < 3𝜋

Let 𝑎 = 𝑥 − 4 and 𝑏 = 𝑦 − 2. The inequality becomes:
𝜋 < |𝑎 + 𝑏𝑖| < 3𝜋

Using the polar representation of complex numbers, 𝑎 + 𝑏𝑖 can be written as:
𝑎 + 𝑏𝑖 = 𝑟(𝑐𝑜𝑠(𝜃) + 𝑖𝑠𝑖𝑛(𝜃))

where 𝑟 = |𝑎 + 𝑏𝑖| is the magnitude of 𝑎 + 𝑏𝑖 and 𝜃 is the argument of 𝑎 + 𝑏𝑖.

We can rewrite the inequality as:
𝜋 < |𝑟(𝑐𝑜𝑠(𝜃) + 𝑖𝑠𝑖𝑛(𝜃))| < 3𝜋

Since 𝑟 is always nonnegative, we can remove the absolute value and rewrite the inequality as:
𝜋 < 𝑟 < 3𝜋

Therefore, the set of points formed by the expression 𝜋 < |𝑧−4+2𝑖| < 3𝜋 is the set of all complex numbers 𝑧 such that the magnitude of 𝑧 − (4 + 2𝑖) is between 𝜋 and 3𝜋.

Answer: The set of points formed by the expression 𝜋 < |𝑧−4+2𝑖| < 3𝜋 is 𝜋 < 𝑟 < 3𝜋, where 𝑧 = 𝑥 + 𝑦𝑖 and 𝑟 is the magnitude of 𝑧 − (4 + 2𝑖).