Transcript TRIGONOMETRY: ONTARIO CURRICULUM vs HISTORICAL

TRIGONOMETRY:
ONTARIO CURRICULUM
vs
HISTORICAL DEVELOPMENT
Carol Miron
TRIGONOMETRY IN
ONTARIO CURRICULUM
Grade 10 (Academic)
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Find lengths and angles of triangles
Trigonometry as ratio of sides
Sine and Cosine Law
Grade 11 (M/U)
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Review of triangle trigonometry
Transformations of sine & cosine
functions
WHY?
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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
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navigation by stars
distances to
distant objects
(parallax)
ARISTOTELIAN TRIGONOMETRY:
Static Applications
Values of sine, cosine, tangent as the
ratio of lengths (right triangles)
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trig tables used in calculations for angles
and lengths
sine and cosine laws for general triangles
Language of mathematics at the time
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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

trigonometric
values not ratios
but entities
P
theta
O
Sine
Cosecant
Cosine
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
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algebra and function notation (Euler)
coordinate system (Descartes)
Static vs Motion Trigonometry
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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
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x   n    1.317i, n  Z
2
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