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.