Sine and cosine are defined as the vertical and horizontal coordinates of the point reached after rotating a given angle counterclockwise from (1, 0) along the circle of radius one centered at the origin. This definition extends the right-triangle ratios to angles of any size, positive or negative, fixes the sign of each function in each quadrant, and makes the outputs periodic with period one full revolution because the same point is reached again after each full turn. It operates in Euclidean plane analytic geometry and is the mapping from angular displacement to Cartesian coordinates that later work in trigonometry and oscillation modelling assumes.
The core principle is the generalization of trigonometric functions (sine, cosine, tangent) from acute right triangles to all real angles via coordinates on a unit circle in Euclidean geometry. Forma…