三角函数

三角恒等式¹

$\sum_{k=1}^{n}k\sin k\alpha=\frac{(n+1)\sin n\alpha-n\sin\left((n+1)\alpha\right)}{4\sin^{2}\frac{\alpha}{2}};$ $\sum_{k=1}^{n}k\cos k\alpha=\frac{(n+1)\cos n\alpha-n\cos\left((n+1)\alpha\right)-1}{4\sin^{2}\frac{\alpha}{2}};$ $\sum_{k=1}^{n}\sin\left[\alpha+(k-1)d\right]=\frac{\sin\frac{nd}{2}\sin\left(\alpha+\frac{n-1}{2}d\right)}{\sin\frac{d}{2}},\quad d\ne0;$ $\sum_{k=1}^{n}\cos\left[\alpha+(k-1)d\right]=\frac{\sin\frac{nd}{2}\cos\left(\alpha+\frac{n-1}{2}d\right)}{\sin\frac{d}{2}},\quad d\ne0;$ $\sum_{k=1}^{n}\sin k\alpha=\frac{\sin\frac{n\alpha}{2}\cdot\sin\frac{n+1}{2}\alpha}{\sin\frac{\alpha}{2}};$ $\sum_{k=1}^{n}\cos k\alpha=\frac{\sin\frac{n\alpha}{2}\cdot\cos\frac{n+1}{2}\alpha}{\sin\frac{\alpha}{2}};$ $\sum_{k=1}^{n}\sin\left(2k-1\right)\alpha=\frac{1-\cos\left(2n\alpha\right)}{2\sin\alpha};$ $\sum_{k=1}^{n}\cos\left(2k-1\right)\alpha=\frac{\sin2n\alpha}{2\sin\alpha};$ $\prod_{k=1}^{n}\cos\frac{k\pi}{2n+1}=\frac{1}{2^{n}};$ $\prod_{k=1}^{n}\sin\frac{k\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^{n}};$ $\prod_{k=1}^{n}\tan\frac{k\pi}{2n+1}=\sqrt{2n+1}.$