Mathematics Faculty

Title of Task: Year 11 …………………………….

Date Task Issued: ………………………

Date Task is Due: week 5term 3 Weighting of Task: 15 %

Head Teacher: Miss S Dawson

Outcomes Assessed: (from the Extension 1 Mathematics Syllabus)
  • PE1appreciates the role of mathematics in the solution of practical problems
  • PE2uses multi-step deductive reasoning in a variety of contexts
  • PE6makes 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 / Grade / Description
14 - 17 / A
Outstanding / The student has demonstrated excellent use of mathematical techniques to obtain insight to practical problems. The geogebra file givesan 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.
10 - 13 / B
High / 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.
7 - 10 / C
Sound / 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.
4 - 6 / D
Basic / 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.
1 - 3 / E
Limited / 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.
0 / N/A / No evidence of knowledge and understanding.

Outcome PE2 Part 1 Marks available: 5.

Mark / Grade / Description
5 / A
Outstanding / The student has demonstrated a perfect, logical argument that demonstrates the correctness of the identity using deductive reasoning
4 / B
High / The student has demonstrated a logical argument that almost demonstrates the correctness of the identity using deductive reasoning
3 / C
Sound / The student has demonstrated some logical arguments that if continued, would demonstrate the correctness of the identity, but it is not complete.
2 / D
Basic / 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.
1 / E
Limited / 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.
0 / N/A / No evidence of deductive reasoning.

Outcome PE6 Part 3 Marks available: 21

Mark / Grade / Description
17 - 21 / A
Outstanding / 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
13 - 16 / B
High / 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
9 - 12 / C
Sound / 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
4 - 8 / D
Basic / 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
1 - 3 / E
Limited / 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
0 / N/A / No evidence of knowledge and understanding.

Comments: (what the student has done well, areas of weakness, what the student can do to improve) ______

______

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

Part1 (5 marks)

Prove.

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 / Description / Picture
  • 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
/
  • Succinctly describe what is happening to the graph, including period, amplitude and shape
/
  • 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.