TRIGONOMETRY: ONTARIO CURRICULUM vs HISTORICAL DEVELOPMENT

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TRIGONOMETRYONTARIO CURRICULUMvsHISTORICAL
DEVELOPMENT

• Carol Miron

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TRIGONOMETRY INONTARIO CURRICULUM

• Grade 10 (Academic)
• Find lengths and angles of triangles
• Trigonometry as ratio of sides
• Sine and Cosine Law
• Grade 11 (M/U)
• Review of triangle trigonometry
• Transformations of sine cosine functions

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WHY?

• Gap from grade 10 to grade 11 in high school
  mathematics
• Examine development of these aspects of
  trigonometry in history and how they relate to
  student learning

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ARISTOTELIAN TRIGONOMETRYStatic Applications

• Trigonometry used to find angles and
  lengths/distances
• heights of buildings, trees(similar triangles)

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ARISTOTELIAN TRIGONOMETRYStatic Applications

• Trigonometry used to find angles and
  lengths/distances
• navigation by stars
• distances todistant objects(parallax)

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ARISTOTELIAN TRIGONOMETRYStatic Applications

• Values of sine, cosine, tangent as the ratio of
  lengths (right triangles)
• trig tables used in calculations for angles and
  lengths
• sine and cosine laws for general triangles
• Language of mathematics at the time
• descriptive problems and proofs
• use of algebraic symbols appearing latter

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ARISTOTELIAN TRIGONOMETRYApplications in Motion

• Circle of radius 1 unit of choice as
  trigonometric values are lengths
• trigonometric values not ratios but entities

Sine Cosecant Cosine Secant Tangent Cotangent
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ARISTOTELIAN TRIGONOMETRYApplications in Motion

• Study behaviour of trigonometric values as angle
  varies
• periodic phenomenon especially in mechanics and
  motion

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ARISTOTELIAN TRIGONOMETRYApplications in Motion

• Needed development of
• algebra and function notation (Euler)
• coordinate system (Descartes)
• Static vs Motion Trigonometry
• is this conceptual difference understood by
  students?
• do students need to unlearn static trigonometry
  to proceed?

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PLATONIC TRIGONOMETRY

• Unifying equation of logarithms, trigonometry,
  complex numbers (Cotes, De Moivre, Euler)

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PLATONIC TRIGONOMETRY

• No real solutions.
• Infinitely many complex solutions