This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley–Klein–Dickson family of spaces of complex Hermitian-type and rank-1. The complex Hermitian elliptic P^N ≡ SU(N + 1)/(U(1) ⊗ SU(N)) and hyperbolic H^N ≡ SU(N, 1)/(U(1) ⊗ SU(N)) spaces, their analogues with indefinite Hermitian metric SU(p + 1, q)/(U(1) ⊗ SU(p, q)) and the non-compact symmetric space SL(N + 1, )/(SO(1, 1) ⊗ SL(N, )) are the generic members in this family; the remaining spaces are some contractions of the former.The method encapsulates trigonometry for this whole family of spaces into a single basic trigonometric group equation, and has universality and self-duality as its distinctive traits. All previously known results on the trigonometry of P^N and H^N follow as particular cases of our general equations.The following topics are covered rather explicitly: (0) description of the complete Cayley–Klein–Dickson family of rank-1 spaces of complex type, (1) derivation of the single basic group trigonometric equation, (2) translation to the basic complex Hermitian cosine, sine and dual cosine laws, (3) comprehensive exploration of the bestiarium of complex Hermitian trigonometric equations, (4) uncovering of a Cartan sector of Hermitian trigonometry, related to triangle symplectic area and co-area, (5) existence conditions for a triangle in these spaces as inequalities and (6) restriction to the two special cases of complex collinear and purely real triangles.The physical quantum space of states of any quantum system belongs, as the complex Hermitian space member, to this parametrized family; hence its trigonometry appears as a rather particular case of the equations we obtain.