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It hasn't been spelt out how an error in 2022 trig paper made its way into the final version of the paper. Alex Welte takes a step by step look at what could have been.
With the department of basic education releasing the national matric results on Thursday, the talk has once again focused on the impossible maths question – question 5.1 in the trig paper – about which there was quite some consternation at the time the exam was written.
Subject experts have publicly confirmed, with detailed analysis, that there was indeed an error in the posing of the question.
What has not been spelt out, probably for reasons of politeness on the part of some, though plausibly more because of cluelessness on the part of those more directly responsible, is how this error made its way into the final version of the paper.
Even this week, we read more empty words about efforts to look into it and prevent further occurrences of such lapses. It's actually awfully simple and painful in an all too familiar way within our faltering education department hierarchy.
Here's the short version:
- Someone (a person or group) came up with a quite sensible question probing basic understanding of trigonometry.
- Someone (quite possibly the same person(s) – but we may never know) then had the totally crazy idea to dress the question up with a bit of indirection – perhaps just to make it more spicy and cute.
- Someone performed the proposed dressing up of the question, and with all the time in the world to get it right, they screwed up the steps which the candidates, while working under time pressure, were supposed to reverse – steps that the candidates would have had to perform right at the beginning of the question, to even see the core problem fully posed in front of them.
- No one managed to check the fancy dressing-up steps to ensure they had been done correctly.
In a properly functioning education department – this could simply not have happened. The core question would have just been presented in a straightforward way, and there would have been no need to check on the correctness of some irrelevant intermediate steps that were spuriously inserted into the process.
Clearly, no one associated with the setting of this question should have such a responsibility, and no one supervising this process should be confused about what is going on, rambling on about investigating what happened.
But to recap - there was a perfectly sensible question, probing several core aspects of basic trigonometry.
Let's look, step by step, at what could have been
When we begin to read the question, we see:
So we know that we need to solve a trigonometric equation – an equation to which the solution is the value of some angle. For the moment, we can put the details of the equation to one side because there are always multiple 'solutions' to such equations unless we explicitly narrow the possibilities somehow. Fortunately, just by reading on to the end of the first line of the question, we see:
We now see that we are looking for an angle between 0 and 90 degrees which will satisfy the equation (to which we will get shortly). Note that it's very normal to use the Greek symbol theta, to label angles in geometry and trigonometry. In standard trigonometry speak, an angle between 0 and 90 degrees can be visualised as a direction, seen from the origin of a 'cartesian plane', pointing in the 'first quadrant,' i.e. the region where both the x and the y coordinates are positive:
It is a very solidly established convention to label the horizontal axis as the x axis (positive values to the right, negative to the left), and the vertical axis as the y axis (positive values up, negative values down) and to think of the defining angle as being measured anticlockwise from the positive x axis.
Looking more closely at the particulars of the equation which we've glossed over, we see that:
Someone sitting for matric exams should know what the sine function (usually abbreviated to just sin) means, and so be able to now draw this picture:
In diagrams like this, it is common to label the 'hypotenuse' with the label R, for radius. It is useful to visualise a circle centred on the origin of a cartesian plane, and to think of indicating the various points on this circle by various choices of the angle theta.
Now the most important theorem of all, in all of school geometry and trigonometry, is the famous theorem of Pythagoras, which explains how the sides of a right-angled triangle are related:
For the set up here, we get:
So now we know that we can understand the mystery angle theta as the one that makes triangles with these proportions:
We can use a calculator to figure out that the angle theta is very close to 56.3 degrees, but the question goes on by quite clearly stating:
Now the first question here is asking – given what we know about the angle theta, what is sine(360° + theta)? It’s pretty foundational to know that when you add 360 degrees to the input of any core trig function which you are evaluating, you will get the same result as you had before you added the extra 360 degrees, so in this case, given that theta was defined by the equation:
It is fair to ask the candidates to come up with the answer that:
Next, we are asked to evaluate tan(theta). A matric learner taking core maths should know what is meant by tan(theta) , which is variously stated as 'opposite over adjacent' or 'Y/X' and so be able to read off our previous diagram
that
Finally we are asked to evaluate cos(180° + theta). Now we have to know how to work with these 'quadrants’ of the cartesian plane and be aware that when we say (180° + theta) we are talking about this:
We also need to know that cosine (usually abbreviated to just cos) is defined, in cartesian coordinate-speak, as ‘X/R’, taking into account that now the x coordinate is negative.
So the answer is
This, which was pretty much the intended process, is actually a decent question for an exam like this. It forces the learner to engage with a number of important central ideas in trigonometry.
What went wrong?
Firstly, the defining equation was presented by calling the mystery angle by the name x. When working in a cartesian coordinate system in the way that is appropriate here, it is 100% conventional to use x and y to label the axes, as we have done here. It makes no sense to begin this question by naming the angle x, instead of using a much more conventional name like theta or some other Greek letter commonly used to name angles. While it is perfectly possible to proceed – there is just no valid reason to interfere with standard notation – especially when it messes with people who are trying to perform under exam pressure.
Secondly (AND MOST CRUCIALLY), the intended defining equation, which, to recap, was
was apparently intended to be presented as
which is not a very big deal, but involved some little rearrangement. Note the minus sign. It is well known to anyone who does mathematics that we all make errors from time to time – actually, quite frequently. When you re-arrange the elements of an equation, you might well accidentally lose track of a minus sign here and there. Perhaps you might arrange the defining equation to this form:
i.e. forgetting the minus sign. The middle of a trig exam is decidedly NOT the time to make a multi-part question hinge entirely on avoiding a little 'sign-error’ in the first step – so this was a terrible idea, to begin with. But to crown it all – the people setting the exam failed at the very sign error challenge which they threw in the path of the exam candidates - and this despite the fact that the examiners had all the time in the world to get it right - and so what they presented to the examinees was precisely this:
Which, if you do it correctly, re-arranges to:
(i.e. unless, following in the footsteps of the examiners, you repeat the sign error built into the question.)
If we now rewind to higher up, we will see that this means we are looking for an angle, in the 'first quadrant' which has a negative value of the sine function. There is no such angle. In fact, there is no angle in the 'first quadrant' which gives a negative value for any of the core trig functions.
Anyone who is really confident in their trigonometry – which we would hope the examiners should be – will see this inconsistency about five seconds into the question.
- Alex Welte is a physicist by training and a jack of all mathematical trades, notably epidemiology
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