Rewrite the following complex number in rectangular form.


z=8(cos⁡(5π4)+isin⁡(5π4))*z*=8(cos(45*π*​)+*i*sin(45*π*​))

Let's break down the given expression step by step.

We have:

\[
z = 8 \left( \cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right) \right)
\]

and

\[
z^* = 8 \left( \cos(45\pi) + i \sin(45\pi) \right)
\]

### 1. First, simplify \( z \):

The angle \( \frac{5\pi}{4} \) is located in the third quadrant of the unit circle, where both sine and cosine are negative.

\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
\[
\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]

Thus, \( z \) becomes:

\[
z = 8 \left( -\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right) \right)
\]
\[
z = 8 \times -\frac{\sqrt{2}}{2} + 8i \times -\frac{\sqrt{2}}{2}
\]
\[
z = -4\sqrt{2} - 4i\sqrt{2}
\]

### 2. Now, simplify \( z^* \):

The angle \( 45\pi \) is coterminal with \( \pi \), since \( 45\pi \mod 2\pi = \pi \). 

For \( \pi \):

\[
\cos(\pi) = -1, \quad \sin(\pi) = 0
\]

Thus, \( z^* \) becomes:

\[
z^* = 8(-1 + i \cdot 0)
\]
\[
z^* = -8
\]

### 3. Final expression in rectangular form:

\[
z = -4\sqrt{2} - 4i\sqrt{2}, \quad z^* = -8
\]

If you want further details, feel free to ask!

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Here are 5 related questions:
1. What is the geometric interpretation of a complex number in polar form?
2. How do you convert between rectangular and polar forms of complex numbers?
3. What are the properties of complex conjugates?
4. Why is \( z^* \) based on \( 45\pi \) simplified to \( \pi \)?
5. What is the significance of the unit circle when dealing with complex numbers?

**Tip**: When finding coterminal angles, subtract multiples of \( 2\pi \) to reduce the angle to one full rotation or less.

Rewrite the following complex number in rectangular form.
z = 8(cos⁡(5π/4) + i sin⁡(5π/4)) * z* = 8(cos(45π) + i sin(45π))

Learn how to convert the complex number z = 8(cos(5π/4) + i sin(5π/4)) into its rectangular form. This problem involves key concepts such as complex numbers in polar form, trigonometric identities, and coterminal angles. Ideal for college-level or advanced high school students.

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