Algebra
# Polar Coordinates

Let $z_1$ and $z_2$ be complex numbers such that $z_1 = 15 \left( \cos \frac{5}{12}\pi + i \sin \frac{5}{12}\pi \right),$ $z_2 = 2 \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right).$ The product $z_1 z_2$ can be expressed as $a + bi,$ where $a$ and $b$ are real numbers. What is the value of $a+b?$

**Details and assumptions**

$i$ is the imaginary number satisfying $i^2 = -1$.

Let $z_1$ and $z_2$ be complex numbers such that $z_1 = 10 \left( \cos \frac{1}{24}\pi + i \sin \frac{1}{24}\pi \right),$ $z_2 = 5 \sqrt{2} \left( \cos \frac{5}{24}\pi + i \sin \frac{5}{24}\pi \right).$ The product $z_1 z_2$ can be expressed as $a + bi,$ where $a$ and $b$ are real numbers. What is the value of $a+b$?

**Details and assumptions**

$i$ is the imaginary number that satisfies $i^2 = -1$.