Download Right angled Trigonometry

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Right angled Trigonometry | Stage 5.3
Summary of Substrands
Duration: 6 weeks
S 5.1 5.2 -Trigonometry
Start Date:
Completion Date:
Teacher and Class:
Outcomes
Mathematics K-10
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MA5.3-1WM uses and interprets formal definitions and generalisations when explaining solutions and or conjectures
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MA5.3-2WM generalisations mathematical ideas and techniques to analyse and solve problems efficiently
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MA5.3-3WM uses deductive reasoning in presenting arguments and formal proofs.
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MA5.3-15MG applies Pythagoras’ theorem trigonometric relationships the sine rule, the cosine rule and the area rule to solve problems, including problems
involving three dimensions.
1
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Overview
Key Words
Suggested Assessment
Identify the sides of triangles as opposite
adjacent hypotenuse. Find the length of sides
and size of angles. Note the language we are
expected to use Tan q
Opposite hypotenuse adjacent degrees minutes seconds
elevation depression curves right angled
Find the height of the flagpole or tree using
trigonometry
In Stage 5, students are expected to know and use the sine,
cosine and tangent ratios. The reciprocal ratios, cosecant,
secant and cotangent, are introduced in selected courses in
Stage 6.
Find the angle of the stairs
Emphasis should be placed on correct pronunciation of 'sin' as
'sine'.
Initially, students should write the ratio of sides for each of the
trigonometric ratios in words, eg
tanθ=side opposite angle θ
side adjacent to angle θ
Abbreviations can be used once students are more familiar
with the trigonometric ratios.
When expressing fractions in English, the numerator is said
first, followed by the denominator. However, in many Asian
languages (eg Chinese, Japanese), the opposite is the case:
the denominator is said before the numerator. This may lead
to students from such language backgrounds mistakenly using
the reciprocal of the intended trigonometric ratio.
Students should be explicitly taught the meaning of the
2
Draw the sine curve cosine curve tangent
curve given the equation
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phrases 'angle of elevation' and 'angle of depression'. While
the meaning of 'angle of elevation' may be obvious to many
students, the meaning of 'angle of depression' as the angle
through which a person moves (depresses) their eyes from the
horizontal line of sight to look downwards at the required point
may not be as obvious to some students.
Teachers should explicitly demonstrate how to deconstruct the
large descriptive noun groups frequently associated with
angles of elevation and depression in word problems, eg 'The
angle of depression of a ship 200 metres out to sea from the
top of a cliff is 25°'.
Students may find some of the terminology encountered in
word problems involving trigonometry difficult to interpret,
eg 'base/foot of the mountain', 'directly overhead', 'pitch of a
roof', 'inclination of a ladder'. Teachers should provide
students with a variety of word problems and they should
explain such terms explicitly.
The word 'trigonometry' is derived from two Greek words
meaning 'triangle' and 'measurement'.
The word 'cosine' is derived from the Latin words complementi
sinus, meaning 'complement of sine', so that
cos 40∘ =sin50∘
.
3
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Rego
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Content
Teaching, learning and assessment
Resources
Stage 5.3 - Trigonometry and Pythagoras'
Theorem §
Students will have done all outcomes from 5.1 and 5.2
3D diagrams using google sketch up
Students:
Watch Youtube clip on 3D trig
Apply Pythagoras' theorem and
Go to internet and print off sheet work through 3D questions
http://www.teachmathsinthinking.co.uk/activities/3duncovered.htm
trigonometry to solve three-dimensional
3D problems-sheets
problems in right-angled triangles
http://www.gobookee.net/get_book.ph
p?u=aHR0cDovL3d3dy50dmRzYi5jYS
93ZWJwYWdlcy9tc3VydGkvZmlsZXM
vNS41JTIwLSUyMDNEJTIwVHJpZyU
yMC5wZGYKNS41IC0gM0QgVHJpZy
BXb3JkIFByb2JsZW1z
(ACMMG276)
 solve problems involving the
lengths of the edges and
diagonals of rectangular prisms
Youtube example 3D trig
and other three-dimensional
http://www.youtube.com/watch?v=_oR
H5ov5w3c
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
4
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(Problem Solving)
 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, eg 'From a point, A,
due south of a flagpole 100 metres
tall on level ground, the angle of
elevation of the top of the flagpole
is 35°. The top of the same
flagpole is observed with an angle
of elevation 22° from a point, B,
due east of the flagpole. What is
the distance from A to B?'

check the reasonableness of
answers to trigonometry word
problems in three dimensions
(Problem Solving)
Use the unit circle to define
Students sketch curves by using calculator to find the angles and plot
curves
Students draw up a table from 0 to 360
And graph curves
Discuss the features of the graph period symmetry
Watch videos on sine and cosine curve
trigonometric functions, and graph them,
Sine and cosine curve
http://videos.kightleys.com/Science/M
aths/23131008_CsD3fs/1880848370_
VMGSWd3#!i=1880848370&k=VMGS
Wd3
with and without the use of digital
technologies (ACMMG274)
 prove that the tangent ratio can be
http://www.sophia.org/sine-wave-
5
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expressed as a ratio of the sine
concept
and cosine ratios, ie
Trig in real life
http://prezi.com/nm6ugpalfi0g/reallife-applications-of-trigonometry/
 use the unit circle and digital
technologies to investigate the
sine, cosine and tangent ratios for
(at least)
and sketch
Angles of any magnitude
the results

http://www.youtube.com/watch?v=9S
OSTKxqMTk
compare the features of
trigonometric curves, including
periodicity and symmetry
(Communicating, Reasoning)

describe how the value of
each trigonometric ratio
changes as the angle increases
from 0° to
360° (Communicating)
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recognise that trigonometric
functions can be used to model
natural and physical
phenomena, eg tides, the
motion of a swinging pendulum
(Reasoning)
Look at graphs to establish angles of any magnitude and their
sign
Verify ALL STATIONS TO CENTRAL works for angles
Use calculator to work out the sign
6
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Students practise finding the sign and the acute angle of obtuse
angles by looking at graphs and using calculator
http://www.youtube.com/watch?v=gBdkY
EbaawA
 investigate graphs of the sine,
cosine and tangent functions for
angles of any magnitude, including
negative angles
7
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 use the unit circle or graphs of
trigonometric functions to establish
and use the following relationships
for obtuse angles, where

recognise that if
Students learn triangles and understand the construction of
triangles- talk about set squares
, then
there are two possible values
for , given
(Reasoning)
 determine the angle of inclination,
, of a line on the Cartesian plane
Students do practise on exact values
by establishing and using the
relationship
where
is
the gradient of the line
8
http://www.amsi.org.au/teacher_modules/
pdfs/Further_trigonometry.pdf
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Solve simple trigonometric equations
(ACMMG275)
 determine and use the exact sine,
cosine and tangent ratios for
angles of 30°, 45° and 60°

solve problems in right-angled
triangles using the exact sine,
cosine and tangent ratios for
30°, 45° and 60° (Problem
Solving)
 prove and use the relationships
between the sine and cosine ratios
of complementary angles in right-
Prove the sine rule – Adjustment For a weaker 5.3 class give
formula
Give students a question where they have to find the connecting
side to find the other side In a triangle
Thus they can see why we can use the sine rule to make the job
easier
angled triangles
 determine the possible acute
and/or obtuse angle(s), given a
trigonometric ratio
9
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Power point on Sine and cosine rule
http://mrstevensonmaths.wordpress.com/
2012/03/29/y11-sine-and-cosine-rulepowerpoint-incl-answers-to-worksheet/
Establish the sine, cosine and area rules
for any triangle and solve related
problems (ACMMG273)
 prove the sine rule: In a given
triangle
http://www.mathworksheetsgo.com/sheets
/trigonometry/advanced/law-of-sines-andcosines/worksheet.php
, the ratio of a side to
the sine of the opposite angle is a
constant
Students do examples to practise finding a side and an angle
 use the sine rule to find unknown
sides and angles of a triangle,
Use the cross to find what sides and angles go together
Ensure you use a triangle where students have to find the other
angle because it is not opposite.
including in problems where there
are two possible solutions when
This resource gives you a step by step
way to teach sine and cosine rule with a
practical example
finding an angle

recognise that if given two sides
http://www.nzmaths.co.nz/sites/default/file
s/ImSoSorryCM1.pdf
and a non-included angle, then
10
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two triangles may result, leading
Prove the cosine rule if a weak 5.3 give the formula
to two solutions when the sine
And show why they can not use the sine rule
rule is applied (Reasoning)
Give students practise using the rule to find sides and angles
Write the formula each time so they can learn the formula
Give triangles where students have to identify what method they
would use
Pythagoras
Right angled trig
Sine rule
Cosine rule
11
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http://www.greatmathsteachingideas.com/
2012/03/12/trigonometry-pile-up/
 prove the cosine rule:
For a given triangle
,
 use the cosine rule to find
unknown sides and angles of a
triangle
12
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 prove and use the area rule to find
13
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the area of a triangle:
For a given triangle
,
 select and apply the appropriate
rule to find unknowns in non-rightangled triangles

explain what happens if the
sine, cosine and area rules are
applied in right-angled triangles
(Communicating, Reasoning)
 solve a variety of practical
problems that involve non-rightangled triangles, including
problems where a diagram is not
provided

use appropriate trigonometric
ratios and formulas to solve
two-dimensional problems that
require the use of more than
one triangle, where the diagram
is provided and where a verbal
description is given (Problem
Solving)
14
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This puzzle can be used to find the area of the shape. It is a
great example why you do not round up too early
15
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Adjustment
Assessment
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Adjustment
Assessment
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Adjustment
Assessment
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Adjustment
Assessment
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Adjustment
Assessment
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Adjustment
Assessment
Evaluation
Adjustment
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