DiBeos

One of the primary objects in trigonometry is the angle, a transcendental quan- tity which can typically only be approximated. By instead using the slope at vertices, many new formulae and discoveries can be found, and even computed by hand. This paper presents a reformu- lation of trigonometry based entirely on constructible numbers, using slope instead of angle as the fundamental quantity. This approach not only provides exact solutions where traditional methods require approximation, but also reveals new solvable cases in triangle construction. Most notably, we present the first complete solution to the Inradius-Side-Side (ISS) case, which surprisingly yields cubic equations. This connection to cubic equations provides a natural bridge from constructible numbers to the broader field of algebraic numbers, offering rich opportunities for teaching both mathematics history and number theory. The pedagogical advantages of this approach include stronger connections to early algebra concepts, delayed introduction of transcendental numbers, and natural progression from elementary to advanced mathematical ideas.