WEBVTT

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Hi I'm Fiona and this is Tamua 2021 paper
2 and question 5.

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We're given 5 lines of a mathematical
argument,

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and we're asked on which line is the
first error.

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So let's go through and have a look at
each line in turn until we come to an

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error.

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We know that we will come to an error
because if you do actually perform this

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substitution of substituting π in for X
into this line here,

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then you do get 0 = 4,
which is mathematically nonsensical.

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And so we know that there's an error.

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So let's go through and find it.

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Line A says sine squared X plus Cos
squared X = 1 for all values of X and

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this holds.

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This is true.

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There's no errors here because this is a
trigonometric identity,

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and an identity is something which holds
no matter what values we choose,

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and so line A is fine.

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Line B says therefore cosine X = sqrt 1
minus sine squared X.

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So let's think about where this comes
from.

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This comes from taking this identity here
and rearranging to get Cos squared X = 1

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minus sine squared X and then taking the
square root of both sides gives cosine X

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= sqrt 1 minus sine squared X.

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And this is where the error is introduced.

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And that's because if we think about
cosine X,

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the values which that function can take
range between -1 and one.

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That means that cosine X can have
negative values,

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in which case when we're taking square
root,

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we need to take the positive square root
and the negative square root.

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Whereas this line says that this
statement is true for all values of X,

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and the error is that it's only true for
some values of X.

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So the answer to this question is that
line B is the first line where an error

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is where an error exists.

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

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One other way that we can view this is by
considering the direction of implications

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between each line of this mathematical
argument.

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When we are constructing A mathematical
argument,

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we want to construct it in such a way
where the implication between each line

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goes both ways or has a if and only if
implication.

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Whereas when we look at the movement from
statement A to statement B,

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then the implication only goes One
Direction.

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So A is necessary for B to hold,
but it's not sufficient.

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Let's look at a numerical example.

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So coming over here on the board,
if I had a ^2 = 100,

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then I could say that A = sqrt 100 is a
solution to this,

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or 10 is one of the values that A could
take.

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But A could also take the value of -10,
and so the implication here goes that way

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because in order for this to hold,
we have to have that a ^2 = 100.

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But A can also take the value of -10 as
well.

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So thinking about things in this way just
helps developing our mathematical logic.