# TRIGONOMETRY: ONTARIO CURRICULUM vs HISTORICAL DEVELOPMENT TRIGONOMETRY: ONTARIO CURRICULUM vs HISTORICAL DEVELOPMENT Carol Miron TRIGONOMETRY IN ONTARIO 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 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 ARISTOTELIAN TRIGONOMETRY: Static Applications Trigonometry used to find angles and lengths/distances heights of buildings, trees (similar triangles)

ARISTOTELIAN TRIGONOMETRY: Static Applications Trigonometry used to find angles and lengths/distances navigation by stars distances to distant objects (parallax) ARISTOTELIAN TRIGONOMETRY:

Static 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

ARISTOTELIAN TRIGONOMETRY: Applications in Motion Circle of radius 1 unit of choice as trigonometric values are lengths P theta trigonometric

values not ratios but entities Sine Cosine O Cosecant Secant Tangent Cotangent ARISTOTELIAN TRIGONOMETRY:

Applications in Motion Study behaviour of trigonometric values as angle varies periodic phenomenon especially in mechanics and motion ARISTOTELIAN TRIGONOMETRY: Applications 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? PLATONIC TRIGONOMETRY Unifying equation of logarithms, trigonometry, complex numbers (Cotes, De Moivre, Euler)

For any real x, eix cos x i sin x ei cos i sin i 1 e 0 PLATONIC TRIGONOMETRY Solve sin x 2

No real solutions. Infinitely many complex solutions: 1 x n 1.317i, n Z 2