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 \(i^2 = -1\).