Coefficients of Sine Terms in Convergent Trigonometric Series

Theorem
Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\openint \alpha {\alpha + 2 \pi}$:


 * $\map f x = \dfrac {a_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)$

Then:
 * $\forall n \in \Z_{\ge 0}: b_n = \dfrac 1 \pi \displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x$

Also see

 * Coefficients of Cosine Terms in Convergent Trigonometric Series