Thursday, 15 October 2015

A point of pendentry? Explain why sin(60) = sqrt(3) / 2.

Very recently, I came across this question from a reputable source:

I have a real problem with this question, which may perhaps be perceived as pedantry, but to me is rather a bugbear. (Actually, it wasn't exactly this question, but the idea is equivalent.)

Clearly, the intent of the questioner is to elicit a response involving one of the 'special triangles' - half of an equilateral triangle, some Pythagoras and the definition of sin(x) as opposite / hypotenuse:

It is not the intent of the question I have a problem with, but the specific wording. The 'solution' shown above does not explain why. In fact, I do not believe it is possible to explain why this fact is true, it simply is true. Beyond the arbitrary definition of degrees and the arbitrary symbols used to represent the numbers and functions involved, the relationship in question is a necessary, must-be maths fact - a universal constant, if you will.

It is no more possible to explain why this fact is true than it is to explain why pi equals 3.14159...

It just does. Indeed this question is more philosophical than it is mathematical - why does dividing the circumference of a circle by it's diameter always necessarily yield this particular constant? And then perhaps, could a universe exist with pi equal to (say) 4?

 As I said, I have no problem with the intent of the question, it's just that the questioner did not mean 'explain why', but rather 'demonstrate that'.

Perhaps this is a pedantic point. As long as the intention of the questioner is communicated to the student, they will be able to 'answer' it. Nevertheless, my teeth will remain well-gritted when I see a question like this.

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