Introduction to Trigonometry: Concepts and Functions

This section introduces the foundational theorems and historical context that underpin the study of trigonometry. Understanding these concepts is essential for grasping the logical structure and applications of trigonometric functions in mathematics.

1.0 Fundamental Theorems in Geometry

Several classical theorems form the basis for trigonometric reasoning and problem-solving. These theorems are essential for understanding the properties of triangles and circles, which are central to trigonometry.

• Teorema de Tales (Thales' Theorem): States that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

• Teorema de Pitágoras (Pythagoras' Theorem): In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Equation:

• Teorema de Euclides/Cordas (Euclid's Theorem/Chord Theorem): Relates to the properties of chords in a circle and their intersections, foundational for understanding circular trigonometry.

Example: Using the Pythagorean Theorem, if a right triangle has legs of length 3 and 4, the hypotenuse is .

2.0 Historical Development and Logarithms

The development of trigonometry is closely linked to the history of mathematics, including the invention of logarithms and the study of arithmetic and geometric progressions.

• John Napier (1550-1617): Mathematician who invented logarithms, greatly simplifying calculations in trigonometry.

• "In Principio erat logos": Latin for "In the beginning was the word," referencing the foundational role of logic and language in mathematics.

• Progressão Aritmética (Arithmetic Progression): A sequence of numbers in which the difference between consecutive terms is constant. General term:

• Progressão Geométrica (Geometric Progression): A sequence where each term is found by multiplying the previous term by a constant ratio. General term:

• Propositiarum Arithmetricarum & Geometricarum: Propositions or theorems related to arithmetic and geometric properties, foundational for advanced trigonometric concepts.

Example: In the geometric progression 2, 4, 8, 16, the common ratio is 2.

3.0 Representations in Trigonometry

Trigonometric concepts have been expressed in various forms throughout history, reflecting the evolution of mathematical language and notation.

• Representações Diacrónicas (Diachronic Representations): The study of how mathematical expressions and notations have changed over time.

• Escrita na forma Retórica (Rhetorical Form): Early mathematical statements were written in full sentences, before the development of symbolic notation.

Example: The Pythagorean Theorem was originally stated in words before being written as .

Additional info: The content above is inferred from the syllabus structure and historical references. These foundational topics are essential for understanding the subsequent chapters in a college trigonometry course.