There are six basic functions of Trigonometry are,

• Sin Function
• Cos Function
• Tan Function
• Cosec function
• Sec Function
• Cot Function

The three basic trigonometric functions formulas are,

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks, particularly Hipparchus (2nd century BC), are credited with the development of trigonometry into a coherent mathematical system. However, the study of triangles can be traced back to the ancient Egyptians and Babylonians more than a millennium before Greek mathematicians took up the study.

Key historical points about the development of trigonometry and its formulas include:

Ancient Origins: Early trigonometry was used in the astronomical calculations of ancient civilizations, including the Egyptians and Babylonians, who made accurate measurements of angles and distances for their constructions and astronomical observations. However, these early practices were more about direct measurements and did not involve trigonometric ratios or formulas as understood today.

Greek Contributions: The Greeks, especially Hipparchus, who is often called the "father of trigonometry," made significant contributions. Hipparchus reportedly created the first trigonometric table for solving triangles. Another important figure was Claudius Ptolemy, whose work "Almagest" in the 2nd century AD included a trigonometric table and several applications of trigonometric concepts in astronomy.

Indian and Islamic Advancements: The development of trigonometry was not confined to the Greeks. Indian mathematicians like Aryabhata, Brahmagupta, and Bhaskara II developed early forms of sine and cosine functions. The Islamic scholars translated Greek works and made significant contributions themselves. Al-Khwarizmi and Al-Battani, among others, expanded the understanding of trigonometry with new theories, methods, and tables, which were later transmitted to Europe.

European Developments: During the Renaissance, European mathematicians translated the Arabic texts into Latin, reintroducing Greek and Islamic mathematical works to Western Europe. Mathematicians such as Johannes Kepler, Bartholomeo Pitiscus (who is credited with coining the word "trigonometry"), and Leonhard Euler made substantial contributions. Euler, in particular, was instrumental in advancing the analytical treatment of trigonometric functions, introducing the modern notation for the sine, cosine, and other trigonometric functions.

Modern Trigonometry: The development of calculus and analytical geometry in the 17th and 18th centuries provided new tools for the expansion of trigonometry. The subject became more abstract and comprehensive, covering the relationships between angles and distances in various geometric figures, not just triangles. Today, trigonometry is a crucial part of mathematics and engineering, with applications ranging from the design of mechanical structures to the analysis of periodic functions.

Trigonometry revolves around several key formulas, which involve the ratios of the sides of a right triangle: sine (sin), cosine (cos), and tangent (tan), among others. The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, forms the basis for many trigonometric identities. Over centuries, various trigonometric identities and formulas have been discovered, such as the sine and cosine laws for any triangle, not just right-angled ones, and the sum and difference formulas, which allow the calculation of the sine, cosine, and tangent of the sum or difference of two angles.

Reference: https://www.teachoo.com/9723/1412/Trigonometry-Formulas/category/2-sin-x-sin-y-formula/