trigonomatriaan

Trigonometriaan, or trigonometry, is the branch of mathematics that studies the relationships between the angles and sides of triangles, along with the periodic phenomena those relationships model. Its central ideas are the trigonometric functions, defined for angles and used to relate angle measures to side lengths. The field originated in ancient astronomy and later became essential to geometry, navigation, and calculus.

Core functions include sine, cosine, and tangent. For a right triangle they are ratios of corresponding sides, and they extend to any angle via the unit circle. Sine and cosine give coordinates on the unit circle; tangent is sine divided by cosine. The Pythagorean identity sin^2 x + cos^2 x = 1 expresses a fundamental relation. Angles may be measured in degrees or radians; radians are natural in calculus.

Key theorems and tools are the Law of Sines, Law of Cosines, and angle addition formulas for sine and cosine. Inverse functions (arcsin, arccos, arctan) recover angles from ratios. The graphs of the basic functions are periodic with period 2π, modeling oscillatory behavior.

Applications span physics, engineering, computer graphics, signal processing, navigation, and astronomy. Trigonometry underpins Fourier analysis, trigonometric interpolation, and spherical trigonometry used in geodesy and astronomy.

History: Elements appear in ancient Mesopotamia and Egypt, with later formalization by Greek mathematicians. The modern theory developed alongside analytic geometry and trigonometric tables, and was linked to complex numbers through Euler’s formula, uniting exponential and trigonometric functions.