Cornwall Education Development Service (www.cornwall.gov.uk/cypf/teaching)

National Centre for Excellence in the Teaching of Mathematics ()

‘Differentiating Sine & Cosine’

Mathematical goalsFor students to:

  • Review and practice previous learning of differentiation and knowledge of trigonometric functions from Core modules 1 and 2;
  • Understand how to generate the gradient graphs from graphs of sine and cosine and deduce the corresponding gradient functions, using symmetry properties where possible. To extend this understanding to graphs of sin2x, cos2xand sin 3x, cos 3x and to generalise for asinbx and acosbx.
  • Appreciate that the result of this process when carried out in radians is more elegant than that produced when working in degrees. Furthermore, that the link between the two types of measurement is a simple scale factor of π/180.;
  • Apply and consolidate their knowledge of the above generalisation in order to differentiate (and at times, for themselves, to integrate) a range of functions of the form asinbx and acosbx or linear combinations of these.

Starting points Students will need to be reasonably confident in working with the properties and transformations of sinx and cosx, both in degrees and radians. They will need to be able to work accurately in estimating the value of a gradient read from a tangent to a curve on a graph.

Resources required(For starter activity) One set of ‘treasure hunt’ question cards placed appropriately around the classroom, preferably in advance. Student ‘treasure hunt’ answer sheets – one sheet per pair.

(For main activity) Pre-prepared resource sheets showing graphs of sinx, sin2x and sin3x (radians), cosx, cos2x and cos3x (radians) and sinx, sin2x and sin3x (degrees) all with blank gradient graphs underneath each – sufficient for one set of three per pair.

Scientific calculators (at least one per pair)

Rulers and some rough paper to work on

Extension materials:

Graph plotting software or graphic calculators (optional)

Graph paper, rulers, scientific calculators

Plenary/review materials:

Pre-cut (or, if not, then original supplied with scissors) ‘Tarsia’ jigsaw activity, glue sticks and paper or card at least A3 in size.

Note: the Tarsia software can be obtained from: . A user manual can also be downloaded from this page.

Time needed60 – 90 minutes

Suggested approachBegin the lesson with the ‘treasure hunt’ activity.This activity is most effective with the students working in pairs, although if there are more than 14 students you will probably need extra question cards or some groups of three. Ideally the ten ‘question’ cards provided should be placed randomly around the classroom on walls, doors, tables etc in advance of the lesson. Each card consists of a question together with an answer to a different question which will be on another card. This is a ‘kinaesthetic’ activity in which students are given a starting question (do this by placing a different answer in each of the ‘start’ circles of the answer sheets given to each pair) the answer to which directs them to another card, whose answer they must find and so on. The value in this activity is that it encourages students to move around, to be competitive (in a friendly way!), to talk and discuss and to self-check their answers (or else they won’t know which card is next!) Position yourself near to the more challenging cards so as to be able to support as needed, but circulate to eavesdrop and encourage as well. Round off the activity once most pairs have finished or are close to doing so. Take brief feedback on issues that arose during the activity.

For the main activity, assign the letters A, B or C to each student, so that you have more or less equal numbers of each. Then ask the As to work with another A, the Bs with another B, and the Cs with another C. You should have mainly pairs, but if you have the occasional three this is fine. Give the set of three sine (in radians) graphs to the As, the three cosine (in radians) graphs to the Bs and the set of three sine (in degrees) graphs to the Cs. You may find it helpful to ask the students to amend the horizontal axes of all the graphs to include some more commonly used values. Now ask the students to draw tangents and make appropriate measurements and calculations in order to be able to plot values of the gradient on the blank graph paper provided on each sheet. Of course, for the 2x and 3x functions, students will need to be careful with their choice of y-axis numbering! As a brief extension, you might, for example, ask groups to draw appropriate graphs to be able to convince themselves about the gradient function of y=½sinx or y=sin(x-π/3).

Allow groups/pairs around 15-20minutes to discuss and tackle the problem, but allow students every opportunity to deduce their three gradient functions. During this time, you should circulate around the groups, encouraging, prompting and posing questions (but not answers!) to each.Make it clear that each pair has to be prepared to feedback what they have discovered to the rest of the class at the end of the activity. Some points you may wish to highlight during discussions might include:

How might the process of obtaining gradients be completed more quickly? (by using symmetry)

What does the periodic nature of the original functions tell you about their gradient functions?

Do the stationary points (or turning points) help?

The sinx (or cosx) graph has a range between -1 and 1 – what about the gradient graph?

If I’d given you the graph of y=2sinx what difference would that have made? What about y=cos(x+π/4)? Convince me!

Explain to me what affect the ‘2’ in y=sin2x has on the corresponding gradient graph?

At a convenient point, draw the class back together so that the As, Bs and Cs can feedback their findings and reflect on the overall conclusions they have come to so far.

Reviewing learningTo review their learning, the ‘Tarsia’ jigsaw is an ideal activity for the students to secure and practice what they have learned in a satisfying way. They should be encouraged to annotate on and around their completed jigsaw by identifying rules and strategies they have used. In this activity, the students must match a function with its derivative edge to edge and assemble the complete jigsaw. Of course, some students may do this by integrating the derivative to obtain the original function! End the lesson by encouraging the students to ‘showcase’ their jigsaws and ensure that all are able to confidently obtain the derivatives of linear sine and cosine functions.

Extending learningEncourage the students to deduce for themselves the generalised approach to integrating asinbx and acosbx.To further extend the activity, you may wish to pose the problem of the derivative of atanbx, which might usefully be done by asking the students to investigate the gradient graphs of various graphs of tanx of their choosing. What will be the implications of the asymptotes of tanx when doing this? Is the gradient ever negative or even zero? What implications will this have for a possible gradient function?

Graph plotting software will be useful to have available in order that students might manipulate various functions for themselves.

A further useful aspect that could be developed is to ask about second (or third, or fourth….) derivatives. What is ‘nice’ about sine and cosine here? Why (by considering the graphs again) might we expect this to happen?

Differentiation lesson – Year 12Cornwall Post-16 Mathematics Project

‘Differentiating Sine & Cosine’Page 1 of 3