# Infinite Trigonometric Sums Part 1

**Geometry**Level 3

The infinite sum below when \(x=\frac{\pi}{6}\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a^{2}+b^{2}\)?

\( \sin x - 2 \sin^{2} x + 3 \sin^{3} x - 4 \sin^{4} x + 5 \sin^{5} x - 6 \sin^{6} x +\ldots \)