We are tasked with finding the modulus of the product of two complex numbers $z_1$ and $z_2$, where the numbers are given in trigonometric (polar) form as:

$z_1 = 3 \left( \cos \frac{\pi}{9} + i \sin \frac{\pi}{9} \right)$ $z_2 = 5 \left( \cos \frac{\pi}{10} + i \sin \frac{\pi}{10} \right)$

Key Steps:

1. Modulus of a Product: The modulus of a product of two complex numbers is the product of their moduli: $|z_1 z_2| = |z_1| \cdot |z_2|$

2. Compute the Moduli:

   • The modulus of $z_1$ is the coefficient before the trigonometric terms, $|z_1| = 3$.
   • Similarly, the modulus of $z_2$ is $|z_2| = 5$.

3. Compute the Result: $|z_1 z_2| = |z_1| \cdot |z_2| = 3 \cdot 5 = 15$

Final Answer:

The modulus of $z_1 z_2$ is 15.

Would you like a more detailed breakdown of the process or related examples? Here are 5 questions to expand your understanding:

1. How is the modulus of a complex number in polar form derived?
2. Why does the argument of a product of complex numbers add in polar form?
3. Can we represent these numbers in Cartesian form and verify the result?
4. How would the solution change if $z_1$ and $z_2$ were given in Cartesian form?
5. What practical applications involve working with complex numbers in polar form?

Tip: Remember, in polar form, multiplying complex numbers simplifies to multiplying their moduli and adding their arguments!