Geometry Concepts and Pythagorean Theorems

Geometry

• "Geo" translates to earth; "metre" means measurement; geometry deals with space properties including distance, shape, and size.
• Key branches of geometry include:
  • Algebraic geometry
  • Discrete geometry
  • Differential geometry
  • Euclidean geometry
  • Non-Euclidean geometry
  • Convex geometry

Egyptian Geometry

• Vladimir Golenishchev was a prominent Russian Egyptologist, known for contributions to the Cairo School of Egyptology.
• The Moscow Mathematical Papyrus is attributed to Golenishchev and includes significant mathematical problems.
• Ahmes, an Egyptian scribe, lived around the end of the Fifteenth Dynasty; he copied the Rhind Mathematical Papyrus (circa 1550 BC).
• The Rhind Mathematical Papyrus contains geometry problems and is a fundamental document in ancient Egyptian mathematics.

Greek Geometry

• Thales of Miletus (6th century BCE) pioneered deductive reasoning in geometry, introducing the proof concept.
• Notable axioms proposed by Thales include:
  • A circle bisected by any diameter creates two equal halves.
  • The base angles of an isosceles triangle are congruent.
  • Angles opposite to crossed lines are equal.
• Pythagoras of Samos (570–495 BC) is known for the Pythagorean theorem, establishing the relationship (A^2 + B^2 = C^2) in right triangles.
• Euclid of Alexandria authored "Elements," laying the groundwork for geometric theorems based on axioms.
• Archimedes of Syracuse (287-212 BC) is renowned for advancing geometric principles, including the area and volume of shapes, utilizing concepts of infinitesimals.

French Geometry

• René Descartes (1596-1650) connected geometry and algebra, creating analytic geometry and standard notations for variables.
• Blaise Pascal (1623-1662) contributed to projective geometry, focusing on properties invariant under projective transformations.
• Conic sections, categorized as hyperbola, parabola, and ellipse, arise from intersecting a cone with a plane, with circles being a special ellipse case.

Trigonometry

• Trigonometry examines sides and angles of triangles; "trigonon" means triangle and "metron" means to measure.
• Early applications of trigonometry involved calculating missing triangle parts based on known values.
• The six trigonometric functions are:
  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Cotangent (cot)
  • Secant (sec)
  • Cosecant (csc)
• Relationships among functions include:
  • (csc A = \frac{1}{sin A})
  • (sec A = \frac{1}{cos A})
  • (cot A = \frac{1}{tan A})

Types and Applications of Trigonometry

• Plane trigonometry deals with angles and distances in two dimensions.
• Spherical trigonometry applies to problems in three-dimensional space.
• Trigonometry is vital in fields like astronomy, navigation, and physical sciences, providing tools for distance and height measurements.

History of Trigonometry

• Classical trigonometry originated in ancient civilizations, notably Egypt and Babylon, which had practical geometry knowledge.
• Babylonian astronomers recorded celestial movements, indicating familiarity with angular measurements.
• The Rhind Mathematical Papyrus reflects early trigonometric concepts related to the "seked," akin to cotangent.
• Hipparchus (c. 190-120 BC), depicted as the "father of trigonometry," created the first trigonometric tables relating angles to chord lengths for astronomy.
• Hipparchus utilized trigonometry to enhance celestial measurements, bridging the gap between mathematical theory and practical application in astronomy.