trigonometrischen

Trigonometrischen is the attributive German form of the adjective trigonometrisch, meaning trigonometric. In mathematics and related sciences, it is used to describe concepts, functions, identities, and methods that relate angles to lengths in triangles and on the unit circle. The form trigonometrischen appears in phrases such as trigonometrischen Funktionen, trigonometrischen Identitäten, and other declined uses within sentences.

In trigonometry, the primary trigonometrische functions are sine, cosine and tangent, defined as ratios in a right triangle: sin θ equals opposite side over hypotenuse, cos θ equals adjacent side over hypotenuse, and tan θ equals opposite side over adjacent. On the unit circle, these definitions extend to all real angles, with periodicity 2π and the fundamental identity sin^2 θ + cos^2 θ = 1. Other functions include cotangent, secant, and cosecant, defined as reciprocals of tan, cos, and sin respectively, where defined. These functions form the core tools for relating angles to lengths and for modeling periodic phenomena.

These functions satisfy numerous identities known as trigonometrische Identitäten, such as sin^2 θ + cos^2 θ = 1, the angle addition formulas, and the relation tan θ = sin θ / cos θ. They underpin analysis in geometry, physics, engineering, and computer graphics, and they play a central role in Fourier analysis when extended to complex numbers via Euler’s formula e^{iθ} = cos θ + i sin θ.

Etymology and history: the term derives from Greek trigono- “three-angled” and metron “measure.” Trigonometric ideas appeared in classical Greece, were developed in Indian and Islamic mathematics, and were later incorporated into European mathematics, where the German form trigonometrischen appears in academic discourse. Related topics include trigonometry, unit circle, trigonometric identities, and Euler’s formula.