Generalizing the Photon-Clock Argument via the Law of Cosines: Full Angular Parametrization, Isotropy of the Lorentz Factor, and the Connection to the Hyperbolic Geometry of Minkowski Spacetime

Context: The standard derivation of the Lorentz factor via the photon‑clock argument uses the Pythagorean theorem, which restricts the reasoning to a mirror oriented perpendicular to the direction of motion. Objective: Generalize the derivation to an arbitrary mirror orientation, and demonstrate that the isotropy of the Lorentz factor is a geometric consequence rather than an assumption. Method: Application of the general law of cosines (Al‑Kashi's theorem) to the triangle formed by the emitter, the mirror, and the moving photon. Results: Separate outbound and return travel times are obtained. Their sum is independent of the angle, proving that the Lorentz factor is isotropic by construction. The coefficients (1 \pm \beta \cos\theta) that emerge are identical to those appearing in the Doppler and aberration formulas. Scope: This work shows that the Euclidean law of cosines is a spatial projection of the null‑interval condition of Minkowski spacetime. The underlying canonical structure is that of hyperbolic rotations. No new physics: This work is a pedagogical and geometric reconstruction; it produces no new physical result.