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We are given two complex numbers in polar (trigonometric) form:

\[
z_1 = 2 \left( \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5} \right)
\]
\[
z_2 = 4 \left( \cos \frac{4\pi}{9} + i \sin \frac{4\pi}{9} \right)
\]

The task is to find the following in trigonometric form:

1. **(a)** The quotient \( \frac{z_1}{z_2} \).
2. **(b)** The product \( z_1 z_2 \).

### (a) Finding \( \frac{z_1}{z_2} \):

In polar form, when dividing two complex numbers \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), the formula is:

\[
\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right)
\]

Here:
- \( r_1 = 2 \), \( \theta_1 = \frac{2\pi}{5} \)
- \( r_2 = 4 \), \( \theta_2 = \frac{4\pi}{9} \)

So:
\[
\frac{r_1}{r_2} = \frac{2}{4} = \frac{1}{2}
\]

Now for the angle:
\[
\theta_1 - \theta_2 = \frac{2\pi}{5} - \frac{4\pi}{9} = \frac{18\pi - 20\pi}{45} = \frac{-2\pi}{45}
\]

Thus:
\[
\frac{z_1}{z_2} = \frac{1}{2} \left( \cos\left( \frac{-2\pi}{45} \right) + i \sin\left( \frac{-2\pi}{45} \right) \right)
\]

### (b) Finding \( z_1 z_2 \):

In polar form, when multiplying two complex numbers \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), the formula is:

\[
z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right)
\]

Here:
- \( r_1 = 2 \), \( \theta_1 = \frac{2\pi}{5} \)
- \( r_2 = 4 \), \( \theta_2 = \frac{4\pi}{9} \)

So:
\[
r_1 r_2 = 2 \times 4 = 8
\]

For the angle:
\[
\theta_1 + \theta_2 = \frac{2\pi}{5} + \frac{4\pi}{9} = \frac{18\pi + 20\pi}{45} = \frac{38\pi}{45}
\]

Thus:
\[
z_1 z_2 = 8 \left( \cos\left( \frac{38\pi}{45} \right) + i \sin\left( \frac{38\pi}{45} \right) \right)
\]

### Final Answers:
(a) 
\[
\frac{z_1}{z_2} = \frac{1}{2} \left( \cos\left( \frac{-2\pi}{45} \right) + i \sin\left( \frac{-2\pi}{45} \right) \right)
\]

(b) 
\[
z_1 z_2 = 8 \left( \cos\left( \frac{38\pi}{45} \right) + i \sin\left( \frac{38\pi}{45} \right) \right)
\]

---

Would you like me to further simplify or explain any steps?

Here are five related questions for deeper understanding:
1. How do you convert between Cartesian and polar form for complex numbers?
2. What is the significance of the argument (angle) of a complex number in polar form?
3. How would the results change if the magnitudes of \( z_1 \) and \( z_2 \) were different?
4. What happens if the angles in a polar representation are negative?
5. How is the product of two complex numbers in polar form geometrically interpreted?

**Tip:** Always simplify the angles to lie between \( 0 \) and \( 2\pi \) for a standard representation in polar form.

Let z_1 = 2 (cos(2π/5) + i sin(2π/5)) and z_2 = 4 (cos(4π/9) + i sin(4π/9)). Find (a) z_1/z_2 and (b) z_1 * z_2 in trigonometric form.

Learn how to solve division and multiplication of complex numbers z_1 and z_2 given in polar form using De Moivre's theorem. The problem involves finding z_1/z_2 and z_1 * z_2 in exact trigonometric form.

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