General Sinus Principles:  Extension of Trigonometric Functions

Abstract Classical trigonometric functions $\sin(\alpha)$, $\cos(\alpha)$, and $\tan(\alpha)$ are implicitly defined under an existential condition: the right-angled triangle, or equivalently, vertical projection. This paper presents a rigorous extension of these functions to a bivariate framework by introducing a second variable, the \emph{projection angle}$\phi \in (0,\pi)$, yielding the generalized functions $\sin(\alpha,\phi)$, $\cos(\alpha,\phi)$, and $\tan(\alpha,\phi)$. denotes the studied angle and $\phi$ the angle of projection. The classical functions are recovered as the special case $\alpha + \phi = \pi/2$. A coherent analytical structure is established, encompassing domains, ranges, symmetry identities, reciprocity relations, summation formulas, and a complete differential and integral calculus. In particular, the $n$-th derivative and $n$-th primitive are expressed in closed form via a rotation operator $\omega(z,n)$. (GST), a product-form decomposition of $\sin(\alpha,\phi)$, $\cos(\alpha,\phi)$, and $\tan(\alpha,\phi)$ as infinite products of their scaled counterparts, together with a polynomial approximation analogous to the Taylor--Young expansion of classical trigonometric functions. The geometric interpretation of all six generalized functions on the composed trigonometric circle is developed, and two- and three-dimensional plots illustrate the analytical properties established. A structural analogy between the surface of the general tangent function $\tan(\alpha,\phi)$ and the fundamental mode of the transient heat equation is identified and supported by a numerical fit to real experimental data ($R^2 = 0.83$), suggesting a potential analytical role for the general trigonometric functions in the theory of parabolic partial differential equations. Mathematics Subject Classification (2010). Primary 26A09; Secondary 51N20, 42A05, 26B05, 44A05.