Trigonometric functions

Definition

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions, including sine, cosine, and tangent, play a crucial role in various mathematical fields, such as geometry and calculus, by allowing the analysis of periodic phenomena, circular motion, and angles in triangles. Their development has significant historical roots, particularly in early Indian mathematics and advances in spherical geometry that expanded their applications in real-world scenarios.

5 Must Know Facts For Your Next Test

1. Indian mathematicians were among the first to develop comprehensive tables for trigonometric functions, which greatly aided in astronomical calculations.
2. Trigonometric functions can be extended to define circular functions, allowing for their use in analyzing periodic behaviors and phenomena.
3. The use of infinite series to represent trigonometric functions was pioneered by mathematicians such as Taylor and Maclaurin, leading to more advanced mathematical techniques.
4. In spherical geometry, trigonometric functions help solve problems related to triangles on spheres, which is essential for navigation and astronomy.
5. The relationships between these functions lead to identities that are foundational for solving various mathematical problems, such as Pythagorean identities and angle addition formulas.

Review Questions

• How did Indian mathematicians contribute to the development and understanding of trigonometric functions?
  • Indian mathematicians made significant contributions by creating extensive tables for trigonometric functions that allowed for more precise calculations in astronomy. This work included defining sine and cosine values for various angles, which facilitated better understanding and application of these functions. Their innovations laid foundational knowledge that later influenced global mathematical practices.
• What are some key advancements in spherical geometry that utilize trigonometric functions, and why are they important?
  • In spherical geometry, trigonometric functions are essential for solving triangles on spheres, a crucial aspect for navigation and astronomy. These advancements allow mathematicians and navigators to calculate distances and angles over the curvature of the Earth, making them vital for accurate mapping and exploration. The application of trigonometric functions in this context illustrates their versatility beyond planar geometry.
• Evaluate how the introduction of infinite series to represent trigonometric functions changed mathematical analysis.
  • The introduction of infinite series representations for trigonometric functions marked a transformative period in mathematical analysis. This development allowed for greater accuracy and flexibility in calculations across various disciplines, such as physics and engineering. By enabling mathematicians to approximate these functions with polynomials through Taylor and Maclaurin series, complex problems could be simplified, leading to new discoveries and insights in calculus and beyond.

Related terms

Sine: A trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

Cosine: A trigonometric function that represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

Tangent:

A trigonometric function that represents the ratio of the sine to the cosine, or equivalently, the ratio of the opposite side to the adjacent side in a right triangle.