<p>This article introduces two new fractional operators with sine and cosine kernels, motivated by the fundamental role of oscillatory signals in electrical engineering and nonlinear electrical circuits. The proposed operators are defined through convolution-type integrals with non-singular trigonometric kernels and admit closed-form Laplace transform representations, enabling direct analytical treatment. Their basic properties are established and their behavior is illustrated through representative examples, including comparisons with the classical Caputo fractional derivative. As an application, a simple memristor model under oscillatory excitation is considered to demonstrate the modeling implications of the proposed operators. In particular, it is shown that the sine-kernel operator yields an exact linear reformulation of the nonlinear memristor model valid for all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, whereas the Caputo-based formulation leads only to locally valid representations. The results highlight the potential of trigonometric-kernel fractional operators for modeling oscillatory systems with memory effects.</p>