Trigonometrie

Trigonometrie, in German-language mathematics, is the branch that studies the relationships between the angles and sides of triangles and, more generally, the relationships between angles and lengths in geometric configurations. It is centered on trigonometric functions and their applications.

The core functions—sine (sin), cosine (cos), and tangent (tan)—are defined for an angle and, in a right triangle, relate the opposite side, adjacent side, and hypotenuse. These functions extend to all real numbers via the unit circle, and their reciprocals—the cosecant (csc), secant (sec), and cotangent (cot)—are defined accordingly. Angles are typically measured in radians in higher mathematics.

The subject includes identities such as the Pythagorean identity sin^2 x + cos^2 x = 1, angle addition formulas, double-angle and half-angle formulas, and the laws of sines and cosines that relate sides and angles in any triangle. Inverse trigonometric functions provide angle measures from ratios, enabling the solution of many geometric and physical problems.

History traces the development of trigonometry from ancient civilizations, with early table-making in Babylon and Greece, significant advances in Indian mathematics and Islamic astronomy, and later formalization in Europe during the Renaissance.

Applications and relevance span geometry, physics, engineering, computer science, signal processing, astronomy, navigation, and computer graphics. Trigonometry models periodic phenomena, rotations, waves, and spatial relationships, and it underpins many numerical methods and algorithms used in science and engineering.

Generalizations connect trigonometry to spherical geometry and to complex analysis via Euler’s formula, linking trigonometric functions with exponential functions and expanding its scope beyond planar contexts.