Gamma Derivatives of the Extended Sine and Cosine Functions for the Upgraded Mass-Spring Oscillatory System

Derivatives in trigonometry have always been defined in orthogonal contexts (i.e., where the y-axis is set perpendicular to the x-axis). Within the context of trigonometric, the present work expands the concept of derivative (operating by the principle of 90 degrees phase shift when applicable to sine and cosine functions) to the realm where the y-axis is at a variable angle $\gamma$ to the x-axis (i.e., non-orthogonal systems). This gives rise to the concept of the \emph{gamma derivative} --- which expands the classical derivative to impart phase shifts of $\gamma$ degrees. Hence, the ordinary derivative (with respect to $\alpha$) or $d/d \alpha$ is a particular case of the more general \emph{gamma derivative} or $d_\gamma/d_\gamma \alpha$. Formula for the $n^{th}$ gamma derivative of the extended sine and cosine functions are defined. For applied mathematics, the gamma derivatives of the extended sine function $\sin^*(\alpha,\gamma)$ and cosine function $\cos^*(\alpha,\gamma)$ determine the extended governing equation of the energy-coupled mass-spring oscillatory system, and by extended analogy that of the electrical LC (Inductance-Capacitance) circuit.