Mathematics Faculty
Title of Task: Year 11
…………………………….
Date Task Issued: ………………………
Date Task is Due: week 5 term 3
Weighting of Task: 15 %
Head Teacher: Miss S Dawson
Outcomes Assessed: (from the Extension 1 Mathematics Syllabus)
 PE1 appreciates the role of mathematics in the solution of practical problems
 PE2 uses multi-step deductive reasoning in a variety of contexts
 PE6 makes comprehensive use of mathematical language, diagrams and notation for
communicating in a wide variety of situations
Topic areas:
 Further trigonometry (sums and differences, t formulae, identities and equations) (Syllabus
topic 5.6–5.9)
Failure to hand this task in by the due date may result in you receiving an N Determination warning letter which clearly
indicates you are at risk of not attaining your Record of School Achievement or Higher School Certificate
Task: Investigate sums of trigonometric functions using formal proofs and geogebra, and
identify situations that model practical situations.
Instructions:
Complete the investigation outlined on the accompanying sheet. You may work with
others, ask for assistance and use any resources you wish. Your solutions must include a
geogebra file that you used to complete the investigation.
Marking Criteria:
Outcome PE1 Part 2, Part 4 Marks available: 17
Mark
14 - 17
Grade
A
Outstanding
10 - 13
B
High
7 - 10
C
Sound
4-6
D
Basic
E
Limited
1-3
0
N/A
Description
The student has demonstrated excellent use of mathematical techniques to obtain insight to practical problems. The
geogebra file gives an accurate model of superposition, and there is logical mathematical reasoning supporting the
evidence about the practically significant values of n. All, or nearly all, arguments are correct with full working
demonstrating mathematical conclusions.
The student has demonstrated thorough use of mathematical techniques to obtain insight to practical problems. The
geogebra file gives a mostly accurate model of superposition, and/or there is some logical mathematical reasoning
supporting the evidence about the practically significant values of n. Most arguments are correct with full working
demonstrating mathematical conclusions.
The student has demonstrated sound use of mathematical techniques to obtain insight to practical problems. The
geogebra file gives a partially accurate model of superposition, and/or there is some logical mathematical reasoning
supporting the evidence about some of the practically significant values of n. Close to half of the arguments are
correct with full working demonstrating mathematical conclusions.
The student has demonstrated basic use of mathematical techniques to obtain insight to practical problems. The
geogebra file does not give an accurate model of superposition, and/or there is little logical mathematical reasoning
supporting the evidence about some of the practically significant values of n.
The student has demonstrated limited use of mathematical techniques to obtain insight to practical problems. The
geogebra file does not give an accurate model of superposition, and/or there is very little logical mathematical
reasoning supporting the evidence about any of the practically significant values of n.
No evidence of knowledge and understanding.
Outcome PE2 Part 1 Marks available: 5.
Mark
5
4
3
2
1
0
Grade
A
Outstanding
B
High
C
Sound
D
Basic
E
Limited
N/A
Description
The student has demonstrated a perfect, logical argument that demonstrates the correctness of the identity
using deductive reasoning
The student has demonstrated a logical argument that almost demonstrates the correctness of the identity
using deductive reasoning
The student has demonstrated some logical arguments that if continued, would demonstrate the correctness
of the identity, but it is not complete.
The student has begun a logical argument towards demonstrating the correctness of the identity, but there are
at least two steps needed to complete the argument.
The student has just begun a logical argument towards demonstrating the correctness of the identity, but there
are at least three steps needed to complete the argument.
No evidence of deductive reasoning.
Outcome PE6 Part 3 Marks available: 21
Mark
17 - 21
13 - 16
9 - 12
4-8
1-3
0
Grade
A
Outstanding
B
High
C
Sound
D
Basic
E
Limited
N/A
Description
The student has demonstrated extensive skill in communicating a description of the shape of the graph, both
visually and using correct mathematical language, for at least 7 values of n
The student has demonstrated thorough skill in communicating a description of the shape of the graph, both
visually and using correct mathematical language, for at least 7 values of n
The student has demonstrated satisfactory skill in communicating a description of the shape of the graph, both
visually and using correct mathematical language, for most of the 7 values of n
The student has demonstrated basic skill in communicating a description of the shape of the graph, visually
and/or using correct mathematical language, for some values of n
The student has demonstrated limited skill in communicating a description of the shape of the graph, visually
and/or using correct mathematical language, for some values of n
No evidence of knowledge and understanding.
Comments: (what the student has done well, areas of weakness, what the student can do to
improve) ______________________________________________________________________
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Investigation
Background Information
Sine and Cosine functions are used by scientists to model waves (light waves, sound waves etc).
Using mathematical models allows scientists and engineers to make predictions about how the
waves will behave.
Shown below are 2 different sine functions (also called sine waves)
In general, when y = Asin(nx), A determines the amplitude, and n determines the period. A higher
n means the peaks of the wave a closer together. The peaks of y=sin3x are 3 times closer together
than y=sinx.
Physicists use the principle of superposition to describe the effect of adding two waves together.
Depending on the relative periods of the two waves, this can result in a resulting wave (or function)
that has a larger ampltitude, a smaller amplitude, or even no amplitude at all (called destructive
interference, where the two waves effectively cancel each other out).
Task
Part 1 (5 marks)
Prove
𝑛𝑥
𝑥
𝑛𝑥
𝑥
sin 𝑥 + sin 𝑛𝑥 = 2sin ( 2 + 2) . cos ( 2 − 2).
Part 2 (2 marks)
Using a slider for n, going from -5 to 5, plot y = sin(x)+ sin(nx) using geogebra. Save the file, with
your name as part of the file name and hand it in with the rest of your solutions
You may watch the video showing this for y = tan(x) + tan(nx) to help you with this. It demonstrates
inserting and using a slider.
Part 3 (21 marks)
Investigate different values of n and describe their effect on the graph, using a table like the one
below.
n
 Use at least 7 different
values of n. You must cover
the full range of n, and
include n= -1, n = 0 and n
=1
Description
 Succinctly describe what is
happening to the graph,
including period, amplitude
and shape
Picture
 Screen shots from
geogebra to illustrate
Part 4 (15 marks)
Using the identity from part 1; investigate mathematically what happens when n= -1, n = 0 and n =
1, and give mathematical reasons why your results correlate with the graph discussed in part 3.