Trigonometry Draft

8/12/2019 Trigonometry Draft

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Trigonometry


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Consider the diagram

What trigonometric functions and identities

can you find in relation to ?


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http://www.gogeometry.com/education/trig

onometry_mind_map.html

http://www.geogebratube.org/book/title/id/8

9516#
http://www.gogeometry.com/education/trigonometry_mind_map.htmlhttp://www.gogeometry.com/education/trigonometry_mind_map.htmlhttp://www.geogebratube.org/book/title/id/89516http://www.geogebratube.org/book/title/id/89516http://www.geogebratube.org/book/title/id/89516http://www.geogebratube.org/book/title/id/89516http://www.gogeometry.com/education/trigonometry_mind_map.htmlhttp://www.gogeometry.com/education/trigonometry_mind_map.html


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History of Trigonometry

Ancient

Sumerian: division of circleinto 360 degrees

Babylonian studies of similartriangles

No systematic method forfinding missing sides etc.

Classical

Euclid andArchimedes and theproperties of chords

Hipparchus,astronomy and thefirst trig table(Almagest: Book onAstronomy andMathematics)

Non western

traditionsAryabhata; Indiantreatments of Sine(6thcentury AD)

Independentdevelopment oftrigonometry in China

Islamic Period

By the 10thCenturythe Greek and Indiantexts had beentranslated

IslamicMathematiciansusing all six trigfunctions and trigtables

Application tospherical geometry


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Contemporary Educational and

Social Context

In school maths: algebraic substitution in

order to achieve computational advantage

mainly in context of analogue computation

(slide rules) and calculus (simplification ofobjects for symbolic manipulation)

In maths, science, computer science, etc.

http://www.gogeometry.com/education/trigonometr

y_mind_map.html


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Problem Solving with Trigonometry

You are a surveyor on a hill opposite a walled cityon a hill on the other side of a shallow valley. Youneed to measure the height of the city wall in orderto inform construction of an aqueduct. You are on

a hundred metre stretch of road going east westand can see the city wall when looking directlyalong the road. You have a clinometer. Come up with a strategy to measure the walls height.

Do you have enough information?

What additional information might you need? Why?


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Challenges in Teaching and

Learning Trigonometry

Discussion of challenges in teaching and learning trig

Prior concepts include a constellation of topics across

geometry, arithmetic, and algebra

Students draw on concepts of ratio given as anumber, manipulation of multiplicative relationships,

and the ability to identify right triangles in atypical

situations

This is often the first time students see functions that

are not polynomials in x and which are represented

using a name. Confusion around notation may be an

issue [ f(x) v sin(x) ]


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Learning TrigonometryNew Concepts and processes

Functions: Although students may have seen polynomials

of x, trig is often the first time they have to attend to

properties of functions that not amenable to simple

algebraic manipulation

Inverse functions

The relationship between function and inverse and the use of

the inverse in symbolic manipulation

Confusion around inverses and Arc-relationship (reciprocalrelationships)?


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Learning Trigonometry

Radians: Geometric principles and implications

and their use in trigonometry Definition; Relationship to measurement in degrees; Reasons

origin Also Tau and the circle constant: The argument for Tau to be

defined as C/r and the illustrative capacity of considering use

of Tau compared to use of Pi.

http://tauday.com/tau-manifesto
http://tauday.com/tau-manifestohttp://tauday.com/tau-manifestohttp://tauday.com/tau-manifestohttp://tauday.com/tau-manifesto


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Learning Trigonometry Non-routine right triangles

Presentation in different orientations

Ambiguous situations

Use of right triangles in problem solving (routineversus non-routine siututaions

Ratios as multipliers

The concept and processes of ratio andproportionality and the application to trigonometric

situations

O t iti d t hi


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Opportunities and teaching

approaches

The opportunity for relational context in which

to address prior misconceptions and

challenges

Ratio concept: right triangles and sine/cos/tan asmulitpliers

Angle as measure of turn by going beyond 90

degrees

O t iti d t hi


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approaches

Teaching approaches Integration of similarity, unit circle and exploratory

approaches as way to address challenges and build onopportunities

Example from personal experience echoes research on how

Trig is often taught: Similarity in right triangles Introduction and definition of SIN/COS/TAN at side lengths ratios

Develop of a trig table (fun!) [this is not as common in the researchon practice; usefule for developing the notion of function andinverse function]

Application in geometry problems

Application in contextualised problem solving Introduction of unit circle radians and graphs of trig function at a

later date

O t iti d t hi


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approaches

New opportunities for development of teaching How does trig understanding develop over time? What

are anticipated learning trajectories and what are theimplications for how to introduce trig and how todevelop it over time?

How can digital technologies be used to enhancestudent understanding of relationship between trig andperiodicity? Motion capture and pendulum problems?

How can trigonometric understanding bedeveloped relationally beyond the secondary

curriculum into issues of fourier transformation,physics, engineering (and statisitics)?


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The use of images in Trigonometric

Problem Solving

Other research

Pritchard etc.


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Standards at A level

2.5 Unit C2 - Core mathematics

4. Trigonometry. The sine and cosine rules, the area of a triangle in the

form 1/2 abSinC. Radian measure, including use for

arc length and area of sector. Sine Cosine and tangent functions. Their graphs,

symmetries and periodicity.

Knowledge and use of tan =sin/cos, and sin2+cos2=1.

Solution of simple trigonometric equations in a giveninterval.


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Standards at A level

2.8 Unit C3Core mathematics.2. Trigonometry. Knowledge of secant, cosecant and cotangent and of

arcsin, arcos and arctan.

Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted

domains.

Knowledge and use of sec2=1+tan2 andcosec2=1+cot2.

Knowledge and use of the double angle formulae; use offormulae for sin (A B), cos(AB) and tan(AB) and ofexpressions for acos+bsinin the equivalent forms ofrcos() or rsin().


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Promoting a Flexible Schema of

Trigonometric Concepts and Processes

Overall research findings indicate that:

The qualitative distinction between individuals trigonometric schemasdepends largely on the focus of the individuals attention when learning.

Students who complemented algebraic processes with spatialrepresentations had a qualitative advantage over those who concentratedtheir attention upon one aspect to the detriment of the other.

The benefit of a schema that had recourse to both spatial and algebraicrepresentations was that:

firstly it was more flexible and;

secondly it could be strengthened on two levels providing greater scopefor understanding.

Challenger, M., 2009. From triangles to a concept: a phenomenographic study of A-level studentsdevelopment of the concept of trigonometry. University of Warwick. Available at:http://wrap.warwick.ac.uk/id/eprint/1935


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Consider the Diagrams from a trigonometric

perspective

What relationships do they represent?

Can you express this algebraically?


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perspective




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Trigonometry Draft

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