trigonometric functions (also called circular functions) are functions of angles; they are important in studying triangles and modeling periodic phenomena and many other applications. trigonometric functions are usually defined as the ratio of two sides of a right triangle containing the angle, or equivalently, as the lengths of various line segments on the unit circle. more modern definitions express them as solutions of infinite series or specific differential equations, allowing them to be extended to arbitrary positive and negative values, even complex values.