WEBVTT

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Hi I'm Fiona and this is TMUA 2021 Paper
one question 6.

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We're told that the function F is given
by F of X equals Cos X + 3 / 7 + 5 Cos X

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minus sine squared X.

c4e48950-e886-49db-8458-1945e9283d14-0
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And we're asked to find the positive
difference between the maximum and

c4e48950-e886-49db-8458-1945e9283d14-1
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minimum values of this function and then
given answers or options A to F to choose

c4e48950-e886-49db-8458-1945e9283d14-2
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from once we get to that point.

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Now for this question,
when we see the words maximum and minimum,

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we might just jump straight to thinking
about differentiation.

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But actually if I take a look at this
function,

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whenever I see cosine X mixed with some
sine squared X's,

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then I'm thinking about this identity Cos
squared X plus sine squared X = 1.

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So I'm first going to use that identity
to get this function in cosine X and then

0863dcf5-5a5c-47f0-9047-fcb7e6edaadf-1
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see where we go from there.

4a414295-25a9-4584-9c95-b9356d7cacfa-0
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So this function can be rewritten to be
Cos X + 3 / 7 + 5 Cos X.

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And then if I rearrange this identity,
I get that sine squared X = 1 minus Cos

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squared X,
so that's -1 minus Cos squared X.

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And just typing up that bottom row,
I have Cos X + 3 all over.

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This seven will have a -1.

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So that's six and +5 Cos X.

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And then this will be minus times minus
plus Cos squared X.

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So now I've got a quadratic in Cos X and
so I'm going to factorise this

00589405-3120-4488-919f-d419813fe59e-1
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denominator.

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So I've got Cos X + 3 all over and then
this denominator will factorise to be Cos

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X.

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Excellent.

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Let me think about this factorization.

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I will have a 3 * 2 to give me that six.

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So +3 and +2 here.

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And now I can see and get a handle on
this function.

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So much more.

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I've got cosine X + 3 on the top and I've
got cosine X + 3 in the denominator.

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And I'm just going to take a moment to
have a look at my function and to check

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that each of the components of my
function are actually all positive.

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That means that there are no no kind of
values of X for which this function is

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undefined.

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And that means that I could actually
cancel this factor of Cos X + 3 with the

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Cos X + 3 on top,
and then I end up with one over Cos X + 2.

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So now I've got this function in a form
that is much easier to see what its

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maximum value is and what its minimum
value is.

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It's a fraction,
and this fraction is going to be at its

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maximum when the denominator is minimised
and at its minimum value when the

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denominator is maximised.

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Now we know that for cosine X and for
sine X as well,

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those are two functions for which the
maximum,

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maximum and minimum values are well known.

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So we know that the minimum value of
cosine X is -1 and so that will give the

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maximum for this function and that will
be when this function is 1 / 2 -,

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1 which is 1.

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So this function has a maximum value of 1
and then it has a minimum value when

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cosine X is maximised.

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So that will be 1 / 1 + 2.

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So that will have a minimum value of 1/3.

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Now I just need to round things up by
taking the positive difference between

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these two values.

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One is 3 thirds.

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Take away 1/3.

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That gives me 2/3 and scanning my options
I can see that option D gives me the

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right value of 2/3.

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So that's the answer to this question.

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Now let's take some time to reflect on
this question.

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And I said at the beginning that when we
see the words maximum and minimum,

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it's tempting to jump straight to
differentiating.

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But if we were to differentiate this
function here,

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especially in its current form,
we would have to use the quotient rule.

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We would have to be using the chain rule
in there as well.

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And then we would get X values.

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And so we'd need to think about the value
of the function based for those X values

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of those turning points.

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This is also a periodic function.

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And so it would just cause a,
a whole load of work.

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And so the thing that made me realise the
approach that I was going to take was

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when I saw this cosine X and sine squared
X.

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And whenever we see that or if we have
sine X and a cosine squared X in,

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then we think about this identity.

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And that should be kind of a tool that we
reach for regularly when we spot these

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kind of things.

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The other thing to mention as well or to
to re emphasise is that sine and cosine

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functions,
we know that they both have a maximum

69783a4c-2e6b-4ffb-81fb-0124438f3ed1-2
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value of 1 and a minimum value of -1.

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And so if I'm asked about maximum and
minimum values of a function containing

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those,
then this kind of approach will be the

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right approach to take because I already
know the maximum and minimum values of

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cosine and sine.