Cornwall Education Development Service (www.cornwall.gov.uk/cypf/teaching)
National Centre for Excellence in the Teaching of Mathematics (www.ncetm.org.uk)
‘Differentiating Sine & Cosine’
Mathematical goals
For 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, cos2x and 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.
Differentiation lesson – Year 12
‘Differentiating Sine & Cosine’
Cornwall Post-16 Mathematics Project
Page 1 of 3
Cornwall Education Development Service (www.cornwall.gov.uk/cypf/teaching)
National Centre for Excellence in the Teaching of Mathematics (www.ncetm.org.uk)
Note: the Tarsia software can be obtained from:
http://www.mmlsoft.com/index.php?option=com_content&t
ask=blogsection&id=15&Itemid=58 . A user manual can
also be downloaded from this page.
Time needed
60 – 90 minutes
Suggested approach
Begin 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 15-20 minutes 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,
Differentiation lesson – Year 12
‘Differentiating Sine & Cosine’
Cornwall Post-16 Mathematics Project
Page 2 of 3
Cornwall Education Development Service (www.cornwall.gov.uk/cypf/teaching)
National Centre for Excellence in the Teaching of Mathematics (www.ncetm.org.uk)
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 learning
To 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 learning
Encourage 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 12
‘Differentiating Sine & Cosine’
Cornwall Post-16 Mathematics Project
Page 3 of 3