WEBVTT

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Hi, I'm Fiona and this is TMUA 2016,
paper one, question 10.

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We're asked 'how many solutions does the
equation XtanX = 1 have in the interval

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between -2π and 2π?'

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And we're given options A to G to choose
from once we get that far.

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So let's dive right into this question.

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I am not asked to find the solutions of
this equation,

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I'm simply asked how many solutions.

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So that tells me I'm going to be able to
take maybe more of a creative approach to

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this question than I'm used to looking at
this equation.

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I am not very familiar with XtanX.

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I might want to take some time to think
about the properties of that function,

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or think about what a sketch of that
function would look like,

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but that's going to take me too much time.

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What I'm going to do is change one thing
about this equation in order to open up

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this question into something that I am
familiar with.

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You might want to take a moment to pause
the video and see if you can spot what

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that is.

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What I'm going to do is divide both sides
by X.

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Now I'm left with an equation which has
functions in it on either side that I am

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familiar with.

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I'm familiar with tanX,
I'm familiar with one over X,

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I just need to take a moment to double
check that I'm OK to divide both sides by

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X by considering what would happen in the
case where X = 0.

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I look back at this equation and when X =
0, I would have 0 = 1,

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so that's not going to be a solution of
this equation.

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And because I'm just considering how many
solutions, I can divide both sides by X,

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and I don't need to worry about what
happens when X = 0.

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Now,
when I'm thinking about solutions and the

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number of solutions to this equation,
then I can think about my graph of tan X,

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my graph of 1 / X,
which is my reciprocal function,

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and how many times they intersect.

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And that will get me my answer.

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So I'm just going to draw a quick sketch
of the graph of tanX between -2π and 2π.

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OK,
here's my graph of tanX and now I'm going

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to overlay this with a graph of my
reciprocal function.

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Now, when I'm drawing these sketches,
I really don't need to be all that

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accurate.

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I just need to be accurate enough so that
I don't miss off any important

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intersections.

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And I can see here that there are 1, 2, 3,
4 intersections of these graphs.

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Looking at my options, that is option E.

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And so that is the answer to this
question.

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Let's take some time to reflect on this
question.

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This is one of those TMUA questions that
the approach is somewhat unfamiliar,

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but I really enjoy these questions
because they cause you to have to really

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lean in to consider what's happening
graphically.

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And in an exam like TMUA,
thinking about what's happening

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graphically,
especially with functions that are

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familiar to you,
can really unlock the question and cause

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you to get into the flow of things.

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Now here,
because we're only asked for how many

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solutions,
a graphical approach is what's needed.

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And so when reflecting back on this
question, I would be thinking,

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I want to be so confident on drawing
really quick sketches of things like tanX,

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sinX and cosX give you a little tip for
sin and cosin X.

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They're obviously very, very similar.

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And if you get mixed up between the two
of them and you're not quite sure which

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ones which,
I always think about the graph of the

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function about the Y axis and the graph
of sine about the Y axis is like an S on

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its side.

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The graph of cosine about the Y axis is
like a C on its side.

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So S for sine, C for cosine.

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And I hope that helps you remember which
ones which.