Với điều kiện \(x\ne k2\pi,\left(k\in\mathbb{Z}\right)\), tổng \(A=sinx+sin2x+sin3x+....+sinnx\) có công thức rút gọn là
\(A=\dfrac{cos\dfrac{x}{2}-cos\dfrac{2n+1}{2}x}{2sin\dfrac{x}{2}}\).\(A=\dfrac{cos\dfrac{x}{2}-cos\dfrac{2n-1}{2}x}{2sin\dfrac{x}{2}}\).\(A=\dfrac{cos\dfrac{x}{2}-cos\dfrac{2n+1}{2}x}{2cos\dfrac{x}{2}}\).\(A=\dfrac{cos\dfrac{x}{2}-cos\dfrac{2n-1}{2}x}{2cos\dfrac{x}{2}}\).Hướng dẫn giải:
\(A.2sin\frac{x}{2}=2sinx.sin\frac{x}{2}+2sin2x.sin\frac{x}{2}+....+2sinnx.sin\frac{x}{2}\)
\(=cos\frac{x}{2}-cos\frac{3x}{2}+cos\frac{3x}{2}-cos\frac{5x}{2}+...+cos\frac{2n-1}{2}-cos\frac{2n+1}{2}x\)
\(=cos\frac{x}{2}-cos\frac{2n+1}{2}x\)
=> \(A=\frac{cos\frac{x}{2}-cos\frac{2n+1}{2}x}{2sin\frac{x}{2}}\)
