Trigonometric functions

Definitions and key properties

There are two complementary viewpoints for defining trigonometric functions. In right-triangle geometry, for an acute angle θ the sine is the ratio of the side opposite θ to the hypotenuse, cosine the adjacent side divided by the hypotenuse, and tangent the opposite divided by the adjacent side. On the unit circle (radius one) each angle corresponds to a point (cos θ, sin θ), which extends definitions to all real angles and connects trig functions to rotational symmetry.

• Periodicity: sine and cosine are periodic with period 2π; tangent has period π.
• Parity: cosine is even (cos(−θ)=cos θ), sine is odd (sin(−θ)=−sin θ).
• Pythagorean identity: sin²θ + cos²θ = 1.
• Calculus relations: derivatives and integrals of trig functions are fundamental in analysis (for example d/dθ sin θ = cos θ).

History and development

Ideas equivalent to trigonometric ratios were used by ancient astronomers and surveyors. Greek mathematicians systematized chord tables; later Indian and Islamic scholars developed sine and cosine in modern form. During the 17th–18th centuries calculus and complex analysis placed trigonometric functions on a firm analytic footing, and Euler's formula linked them to the exponential function. For background on angles and geometric origins see angles and classical treatments such as triangle geometry.

Uses and examples

Trigonometric functions model oscillations, waves and rotations. They appear in navigation and astronomy (bearing and celestial coordinates), in engineering (signal processing, alternating current theory), and in applied mathematics (Fourier series and transforms). Simple examples include representing a vibrating string or decomposing a periodic signal into sine and cosine components; many textbooks and applications illustrate these uses in detail here.

Related functions and distinctions

Besides the six classical functions, inverse trigonometric functions (arcsin, arccos, arctan) return angles from ratios. Hyperbolic functions (sinh, cosh, tanh) are analogous but arise from exponential definitions rather than circular geometry. Extending trig functions to complex arguments reveals deep links to exponentials and analytic continuation; more on these analytic aspects is available in advanced references.