Trigonometric functions

Trigonometric functions assign values such as sine, cosine, and tangent to angles, encoding geometric ratios and extending periodically beyond triangle angles.

Angle: A geometric figure formed by two rays sharing an endpoint, or a measure of rotation between directions. Angles are the inputs whose geometric relationships these functions encode.
Sine: A trigonometric function giving the vertical coordinate of a point on the unit circle at a given angle. It is one of the three principal functions and underlies many identities.
Fourier series: A representation of a periodic function as a sum of sine and cosine terms. It uses trigonometric functions as building blocks for analyzing periodic signals.
Inverse trigonometric functions: Functions that return selected angle values from trigonometric outputs. They reverse trigonometric mappings only after restricting their domains and ranges.
Unit circle: A circle of radius one centered at the origin of a coordinate plane. Coordinates on this circle define sine and cosine for angles beyond triangle ranges.
Cosine: A trigonometric function giving the horizontal coordinate of a point on the unit circle at a given angle. Its phase-shift relation to sine links horizontal and vertical circular motion.
Simple harmonic motion: Periodic motion in which displacement varies sinusoidally with time. Sine and cosine describe ideal oscillations such as a mass on a spring.
Hyperbolic functions: Functions defined using exponential combinations that resemble trigonometric functions but are not periodic. Their identities resemble circular-function identities, while their geometry and behavior differ.
Radian: An angle measure equal to the ratio of an arc's length to its radius. Radians make periodic formulas and calculus identities especially direct.
Tangent: The quotient of sine by cosine, undefined where cosine is zero. Its quotient definition explains its poles and shorter period.

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