To solve for $t$, we need to equate the given complex number form with the expression for $z_2$.

We are given: $z_2 = \frac{3\pi}{4} \quad \text{and} \quad z_2 = t + 8i$

So, we know that the complex number in polar form $z_2 = \frac{3\pi}{4}$ corresponds to an angle of $\frac{3\pi}{4}$ in the complex plane, and we need to express this in rectangular form (i.e., $t + 8i$).

For $z_2 = r(\cos \theta + i \sin \theta)$, we need to identify the real and imaginary components. Here, the modulus $r$ is not given explicitly, but we can deduce it from the imaginary part (8i). The imaginary part is given as $8i$, so the modulus is $r = 8$.

Thus, we can express $z_2$ as: $z_2 = 8\left( \cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)$

Evaluating the trigonometric values: $\cos\frac{3\pi}{4} = -\frac{1}{\sqrt{2}}, \quad \sin\frac{3\pi}{4} = \frac{1}{\sqrt{2}}$

So: $z_2 = 8\left( -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right)$

Now, compare this with $z_2 = t + 8i$: $t + 8i = -8 \frac{1}{\sqrt{2}} + 8i \frac{1}{\sqrt{2}}$

Thus, the real part $t$ is: $t = -8 \frac{1}{\sqrt{2}} = -\frac{8}{\sqrt{2}} = -4\sqrt{2}$

So, the value of $t$ is $-4\sqrt{2}$.

Would you like further details on this?

Here are some related questions to delve into:

1. How to convert between polar and rectangular forms of complex numbers?
2. What does the modulus of a complex number represent geometrically?
3. How to find the argument of a complex number in polar form?
4. How to graph a complex number in the complex plane?
5. What are the properties of complex conjugates?

Tip: To avoid confusion, always write down both real and imaginary parts when working with complex numbers in rectangular form!