The following nice variant of the Fejér-Jackson inequality is proved: For all natural numbers \(n\) and real numbers \(x\in [0,\pi]\) we have \[-0.05781\ldots=\] \[-(5/48)\sqrt{130-58\sqrt{5}}\leq F_{n}(x),\] where \(F_{n}(x)=\sum_{k=1}^{n}\left( -1\right) ^{k+1}\left( \frac {\sin\left( (2k-1)x\right)}{2k-1}+\frac{\sin\left( 2kx\right)}{2k}\right) ;\) the sign of equality holds if and only if \(n=2\) and \(x=4\pi/5\).