Trigonometry: Definition, Formulas, Ratios, & Identities

Key Concepts

• Trigonometry is a branch of mathematics focused on the relationships between angles and sides in triangles.
• It involves six main trigonometric functions:
  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Cotangent (cot)
  • Secant (sec)
  • Cosecant (csc)

Historical Development

• Classical Trigonometry

  • Originated from Greek words "trigonon" (triangle) and "metron" (to measure).
  • Initially focused on calculating missing parts of triangles.
  • Important figures: Hipparchus, who developed a table of chords and spherical triangles.

• Egyptian and Mediterranean Influence

  • The Egyptians used the concept of "seked" related to the slope of pyramids.
  • The Rhind Papyrus contains early instances of trigonometric concepts.

• Indian and Islamic Contributions

  • Aryabhata I developed the first table of sines, using the term "jya".
  • Muslim scholars translated and expanded on Indian works, introducing concepts like tangents and cotangents.

Trigonometric Principles

• Trigonometric Functions

  • Functions are used to find unknown angles and distances in geometric figures.
  • Each function is defined relative to a right triangle.

• Applications

  • Used in astronomy, mapmaking, surveying, and artillery.
  • Plane trigonometry deals with problems in a single plane.
  • Spherical trigonometry addresses issues in three-dimensional space.

Passage to Europe

• Spherical trigonometry was predominant until the 16th century.
• Regiomontanus's "On Triangles of Every Kind" became a key work in trigonometry.
• John Napier's invention of logarithms greatly aided trigonometric calculations.

Key Historical Figures

• Hipparchus: Developed early trigonometric function tables.
• Ptolemy: Advanced trigonometry through his work "Almagest".
• Aryabhata I: Introduced the sine function in India.
• Regiomontanus: Authored the first modern trigonometry book.
• John Napier: Introduced logarithms.

Related Topics

• Concepts of plane and spherical triangles.
• Development of trigonometric tables and their historical significance.