analytic trigonometry

Analytic trigonometry is a branch of trigonometry that uses a coordinate system and algebraic manipulation of trigonometric functions to derive formulas valuable in science and engineering. It defines trigonometric functions for real numbers, not just angles, and these functions satisfy specific relations and have minimum periods.1

Key aspects include:
* Definitions: Trigonometric functions like sine and cosine can be defined using power series, allowing for precise calculations.1
* Inverse Functions: Each trigonometric function has an inverse, such as arcsin or sin⁻¹, which "undoes" the original function.1
* Identities: Various algebraic identities, like the Pythagorean identity and formulas for half-angles and double-angles, are crucial for simplifying expressions and solving equations.2
* Applications: Analytic trigonometry is applied in fields such as surveying, engineering, and navigation to solve problems involving triangles.3

The development of analytic trigonometry in the 16th century was influenced by symbolic algebra and analytic geometry, leading to its integration with algebraic and complex number concepts.4