Summary: This paper is concerned with series of the form \[ \Phi(\theta)=A_\Phi(\theta)+\sin\theta\sum_{k=1}^\infty\varphi_k\sin k\theta, \] where \( \Phi(\theta)\) is an even \( 2\pi\)-periodic function with finite values \( \Phi(0)\) and \( \Phi(\pi)\), { } \[ \begin{aligned} & A_\Phi(\theta)=\frac{\Phi(0)+\Phi(\pi)}{2}+\frac{\Phi(0)-\Phi(\pi)}{2}\cos\theta,\qquad\varphi(\theta)=\Phi(\theta)-A_\Phi(\theta),\\ & \varphi_k=\frac{2}{\pi}\int_0^\pi\varphi(t)\frac{\sin kt}{\sin t}\,dt. \end{aligned} \] Series of this type appear as a particular case of more general special series in ultraspherical Jacobi polynomials, which were first introduced and studied by the author. Partial sums of the form \( \Pi_n(\Phi)=\Pi_n(\Phi,\theta)=A_\Phi(\theta)+\sin\theta\sum_{k=1}^{n-1}\varphi_k\sin k\theta\) are shown to have a number of important properties, which give them an advantage over trigonometric Fourier sums of the form \( S_n(\Phi,\theta)=\frac{a_0}{2}+\sum_{k=1}^na_k\cos k\theta\). Approximation properties of Fejér- and de la Valleé-Poussin-type means for the partial sums \( \Pi_n(\Phi,\theta)\) are studied.