Download Chapter 13: Trigonometry

Chapter 13: Trigonometry
Stages 5.1/5.2/5.3 –
Year 9
Unit length: 4 weeks
Strands:
Measurement and Geometry
Substrands:
Rationale:
Trigonometry is a branch of geometry that is very important in fields such as
navigation, surveying, engineering, astronomy and architecture. Students will use
basic trigonometry to find unknown sides and angles in right-angled triangles.
Teacher:
Outcomes:
Right-Angled Triangles
(Trigonometry); Trigonometry
and Pythagoras’ Theorem
Review of earlier work from Stage 4 and selections from:
Right-Angled Triangles (Trigonometry) [Stages 5.1, 5.2◊],
Trigonometry and Pythagoras’ Theorem [Stage 5.3§]
Dates taught:
A student:
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uses appropriate terminology, diagrams and symbols in mathematical
contexts (MA5.1-1WM)
selects and uses appropriate strategies to solve problems (MA5.1-2WM)
provides reasoning to support conclusions that are appropriate to the context
(MA5.1-3WM)
selects appropriate notations and conventions to communicate mathematical
ideas and solutions (MA5.2-1WM)
interprets mathematical or real-life situations, systematically applying
appropriate strategies to solve problems (MA5.2-2WM)
uses and interprets formal definitions and generalisations when explaining
solutions and/or conjectures (MA5.3-1WM)
generalises mathematical ideas and techniques to analyse and solve
problems efficiently (MA5.3-2WM)
uses deductive reasoning in presenting arguments and formal proofs
(MA5.3-3WM)
applies trigonometry, given diagrams, to solve problems, including problems
involving angles of elevation and depression (MA5.1-10MG)
applies trigonometry to solve problems, including problems involving bearings
(MA5.2-13MG)
applies Pythagoras’ theorem, trigonometric relationships, the sine rule, the
cosine rule and the area rule to solve problems, including problems involving
three dimensions (MA5.3-15MG)
________ to ________
Teacher reflection
Strengths
Weaknesses
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
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Content
statements:
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Use similarity to investigate the constancy of the sine, cosine and tangent
ratios for a given angle in right-angled triangles (ACMMG223)
Apply trigonometry to solve right-angled triangle problems (ACMMG224)
Solve right-angled triangle problems, including those involving direction and
angles of elevation and depression (ACMMG245)
Apply Pythagoras’ theorem and trigonometry to solve three-dimensional
problems in right-angled triangles (ACMMG276)
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Resources:
Literacy:
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Foundation worksheets
Drag-and-drop activities
Interactive lesson
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Other
Challenge worksheets
GeoGebra activities
Videos
adjacent side
bearing
opposite side
tangent ratio
angle of depression
cosine ratio
similar triangles
trigonometric ratio
angle of elevation
hypotenuse
sine ratio
trigonometry
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
2
Student Book / eBook
13:01 Right-angled
triangles
Content dot points
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13:02 Similar right-angled
triangles: The ratio
of sides
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13:03 Trigonometric ratios •
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Register
Technology
identify the hypotenuse, adjacent
sides and opposite sides with
respect to a given angle in a rightangled triangle in any orientation
label sides of right-angled
triangles in different orientations in
relation to a given angle
label the side lengths of a rightangled triangle in relation to a
given angle
Drag-and-drop
define the sine, cosine and
tangent ratios for angles in rightangled triangles
use similar triangles to investigate
the constancy of the sine, cosine
and tangent ratios for a given
angle in right-angled triangles
GeoGebra
use trigonometric notation
find the size in degrees and
minutes of unknown angles in
right-angled triangles
Maths terms 13
Investigating the ratios
of sides of similar rightangled triangles
Resources and suggestions
Review the labelling of sides according to the
angle given. Use triangles in different
orientations.
Use Questions 1 and 2 of Exercise 13:02
(pp. 379–380) to discover the relationship
between the sides of a triangle for a 30 angle
and 50 angle. Use this to generalise the
pattern. Use ‘SOH CAH TOA’ as a helpful
means for remembering these ratios.
Preview the GeoGebra activity on p. 381.
Drag-and-drop
The trigonometric ratios
Videos
The sine ratio –
introduction
The tangent ratio –
introduction
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
Read through pp. 382–383 to understand why
tan  = o/a.
Show students how to draw the commonly
used Greek letters and liken them to other
pronumerals previously used.
Challenge worksheet 13:03 – The range of
values of the trigonometric ratios
Use memory strategies to remember the
relationship between the trigonometric ratios,
e.g. ‘SOH CAH TOA’ or ‘Some Old Hare Came
A Hopping Through Our Area’, p. 384.
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13:04 Trig ratios and the
calculator
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Ensure students have their calculators for this
lesson. Students need to be aware of the
‘degrees and minutes’, ‘sin’, ‘cos’, ‘tan’ and
‘shift’ (or ‘2ndF’) buttons.
use a calculator to find the values
of the trigonometric ratios, given
angles measured in degrees and
minutes
Investigation 13:04 – The exact values for the
trig ratios 30°, 60° and 45° (p. 389)
13:05 Finding an
unknown side
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find the lengths of unknown sides
in right-angled triangles where the
given angle is measured in
degrees and minutes
select and use appropriate
trigonometric ratios in right-angled
triangles to find unknown sides,
including the hypotenuse
Drag-and-drop
Finding sides
GeoGebra
Finding an unknown
side using the tangent
ratio
Foundation worksheet 13:05 – Using
trigonometry to find side lengths
Help students draw diagrams representing the
problems described.
Preview the GeoGebra activities on p. 394.
Finding an unknown
side using the sine and
cosine ratios
Finding the hypotenuse
using the sine and
cosine ratios
13:06 Finding an
unknown angle
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use a calculator to find
approximations of the
trigonometric ratios for a given
angle measured in degrees
use a calculator to find an angle
correct to the nearest degree,
given one of the trigonometric
ratios for the angle
use a calculator to find the size in
degrees and minutes of an angle,
given a trigonometric ratio for the
angle
select and use appropriate
trigonometric ratios in right-angled
triangles to find unknown angles
correct to the nearest degree
Drag-and-drop
Finding angles
GeoGebra
Using the trigonometric
ratios to find an angle
Ensure students know how to use their
calculators when finding unknown angles.
Challenge worksheet 13:06 – Trigonometry
and the limit of an area
Preview the GeoGebra activity on p. 397.
Interactive lesson
Trigonometric ratios in a
right-angled triangle
Videos
The sine ratio – finding
the angle (wheelchair
ramps)
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
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13:07 Miscellaneous
exercises
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solve a variety of practical
problems involving angles of
elevation and depression,
including problems for which a
diagram is not provided
draw diagrams to assist in solving
practical problems involving
angles of elevation and
depression
interpret three-figure bearings
(eg 035°, 225°) and compass
bearings (eg SSW)
interpret directions given as
bearings and represent them in
diagrammatic form
solve a variety of practical
problems involving bearings,
including problems for which a
diagram is not provided
draw diagrams to assist in solving
practical problems involving
bearings
check the reasonableness of
solutions to problems involving
bearings
Drag-and-drop
Bearings 1
Bearing 2
Video
Bearings – an
introduction
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
Introduce new terminology: ‘angle of elevation’
and ‘angle of depression’ (p. 398).
Read p. 399 and explain the two types of
compass bearings that will be used in
trigonometry.
Foundation worksheet 13:07A – Angles of
elevation and depression, and bearings
Foundation worksheet 13:07B – Problems
with more than one triangle
As a class, use compasses to navigate around
the school. Come back to the classroom and
draw the route taken including the relevant
bearings.
Use clinometers to investigate angles of
depression and elevation around the school.
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13:08 Three-dimensional
problems
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solve problems involving the
lengths of the edges and
diagonals of rectangular prisms
and other three-dimensional
objects
use a given diagram to solve
problems involving right-angled
triangles in three dimensions
check the reasonableness of
answers to trigonometry problems
involving right-angled triangles in
three dimensions
draw diagrams and use them to
solve word problems involving
right-angled triangles in three
dimensions, including using
bearings and angles of elevation
or depression
check the reasonableness of
answers to trigonometry word
problems in three dimensions
Review
Use three-dimensional images or solids to help
students visualise right-angled triangles found
in objects.
Ensure students check the reasonableness of
their answers when solving problems.
Make use of Maths terms 13. (p. 406)
Diagnostic test 13 – Use the right-hand
column to assist in remediation when errors
occur. (p. 407)
Assignment 13A – Exam-style questions for
revision (p. 408)
Assignment 13B – Working mathematically
problems (p. 409)
Assignment 13C – Use this cumulative
revision to review previous topics (p. 410)
Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.3 Teaching Program — Chapter 13
Copyright © Pearson Australia 2013 (a division of Pearson Australia Group Pty Ltd) ISBN 978 1 4860 0531 4
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