trigonometric

Trigonometric is an adjective related to trigonometry, the branch of mathematics that studies the relationships between the angles and sides of triangles and the periodic phenomena that arise from those relationships. The term comes from Greek roots meaning “three angles” and “to measure.”

Trigonometric functions are the central tools of trigonometry. The primary functions are sine, cosine, and tangent, along with their reciprocal functions cosecant, secant, and cotangent. These functions can be defined in two equivalent ways: in a right triangle as ratios of side lengths, or on the unit circle in the coordinate plane, where each angle corresponds to a point with coordinates (cos θ, sin θ). This unit-circle view extends the functions to all real angles and clarifies their periodic nature.

Key properties of trigonometric functions include periodicity, with a fundamental period of 2π for sine and cosine, and extended periodic behavior for the other functions. Important identities include the Pythagorean identity sin²θ + cos²θ = 1, reciprocal identities csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ, and angle addition formulas such as sin(a ± b) = sin a cos b ± cos a sin b and cos(a ± b) = cos a cos b ∓ sin a sin b. Inverse functions (arcsin, arccos, arctan) are used to find angles from known ratios.

Applications of trigonometric concepts span science, engineering, computer graphics, acoustics, and physics, where they model waves, oscillations, rotations, and periodic phenomena. The study of trigonometry has a long history, with early contributions from ancient Greek, Indian, and Islamic mathematics, and it matured into a foundational tool in modern mathematics and applied disciplines.