The expression provided is in the form of Euler's formula:

$z = \cos\left(\frac{3\pi}{4}\right) + i \sin\left(\frac{3\pi}{4}\right)$

This corresponds to a complex number in polar form, where:

$z = e^{i\theta}$

Here, $\theta = \frac{3\pi}{4}$. Let's break down the values of $\cos\left(\frac{3\pi}{4}\right)$ and $\sin\left(\frac{3\pi}{4}\right)$:

• $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
• $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$

So the complex number becomes:

$z = -\frac{\sqrt{2}}{2} + i \cdot \frac{\sqrt{2}}{2}$

This is the rectangular form of the complex number.

Would you like further details on polar to rectangular conversion or other properties of this complex number?

Here are 5 questions to consider:

1. How do you convert between polar and rectangular forms of complex numbers?
2. What is the significance of the angle $\frac{3\pi}{4}$ in the unit circle?
3. How does Euler's formula relate to this problem?
4. Can you represent this number in exponential form?
5. What are the applications of complex numbers in polar form?

Tip: For complex numbers, the modulus and argument help provide a clearer geometric understanding on the complex plane.