The problem involves finding the product $z_1 z_2$ and the quotient $\frac{z_1}{z_2}$ for the given complex numbers in polar form.

The given complex numbers are: $z_1 = \cos\left(\frac{7\pi}{8}\right) + i\sin\left(\frac{7\pi}{8}\right),$ $z_2 = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right).$

In polar form, a complex number $z = \cos\theta + i\sin\theta$ is written as $e^{i\theta}$.

Converting to polar exponential form:

$z_1 = e^{i\frac{7\pi}{8}}, \quad z_2 = e^{i\frac{\pi}{4}}.$

1. Finding the product $z_1 z_2$:

The product of two complex numbers in polar form is obtained by adding their angles: $z_1 z_2 = e^{i\frac{7\pi}{8}} \cdot e^{i\frac{\pi}{4}} = e^{i\left(\frac{7\pi}{8} + \frac{\pi}{4}\right)}.$

Simplify the angle: $\frac{7\pi}{8} + \frac{\pi}{4} = \frac{7\pi}{8} + \frac{2\pi}{8} = \frac{9\pi}{8}.$

Thus: $z_1 z_2 = e^{i\frac{9\pi}{8}}.$

In polar form: $z_1 z_2 = \cos\left(\frac{9\pi}{8}\right) + i\sin\left(\frac{9\pi}{8}\right).$

2. Finding the quotient $\frac{z_1}{z_2}$:

The quotient of two complex numbers in polar form is obtained by subtracting their angles: $\frac{z_1}{z_2} = \frac{e^{i\frac{7\pi}{8}}}{e^{i\frac{\pi}{4}}} = e^{i\left(\frac{7\pi}{8} - \frac{\pi}{4}\right)}.$

Simplify the angle: $\frac{7\pi}{8} - \frac{\pi}{4} = \frac{7\pi}{8} - \frac{2\pi}{8} = \frac{5\pi}{8}.$

Thus: $\frac{z_1}{z_2} = e^{i\frac{5\pi}{8}}.$

In polar form: $\frac{z_1}{z_2} = \cos\left(\frac{5\pi}{8}\right) + i\sin\left(\frac{5\pi}{8}\right).$

Final Answers:

1. $z_1 z_2 = \cos\left(\frac{9\pi}{8}\right) + i\sin\left(\frac{9\pi}{8}\right)$.
2. $\frac{z_1}{z_2} = \cos\left(\frac{5\pi}{8}\right) + i\sin\left(\frac{5\pi}{8}\right)$.

Let me know if you would like a detailed step-by-step breakdown of any part!

Here are five related questions:

1. What are the magnitudes of $z_1$ and $z_2$?
2. How do you convert between rectangular and polar forms of complex numbers?
3. Can you plot $z_1$ and $z_2$ on the complex plane?
4. What happens to the angles in polar form when you compute powers of complex numbers?
5. How do you find the conjugate of a complex number in polar form?

Tip: When multiplying or dividing complex numbers in polar form, remember that you only need to work with the angles—magnitudes are multiplied for products and divided for quotients.