I had to fix question 3 of the test in the end. It was bugging me too much. After I'd emailed the replacement to Chris [who replied saying it was fine to hand in electronically, implying he also didn't mind me doing it twice] I went home with the intention of catching up on sleep. This was less successful than it could have been, given that I had to get up twice during the night to check that certain dreams about horrible errors in the test were not in fact true. I should have seen that coming and put a copy on my bedside table before going to sleep.

As it happens there *is* one other error that I've spotted, but I feel fine about that one because I didn't actually understand what I was saying when I said it. The fact that understanding, and subsequent recognition of the error, has come *after* doing the test apparently makes it okay, whereas the question 3 error was just unforgivably careless.

Today has not been productive, and thus I have not been cheerful. I've been fiddling around with geometry, trying to come up with something on SOHCAHTOA to make my student go "wow". When I was 14 I remember being vastly impressed by an exercise which involved comparing the lengths of sides in triangles whose hypotenuses were all radii of the unit circle. Reproducing it now, I just can't see what was so special about it. I don't think I've missed any aspects of the exercise, so I must have forgotten something important about being 14 and not understanding algebra.

The main conclusion of this post seems to be "I'm crusty already".

## 7 comments:

The Gnome.

I'm wondering what your triangles in a circle were. Surely there must have been a little more to it than merely the hypotenuse being unit length. Such as the other point being on the circle maybe?

Are you sure it wasn't the diameter of a unit circle? And the other point on the circle, in which case the other angle is always a right angle?

The one about lines drawn from a fixed point to intersect a circle, and the product of the distances from the point to the intersections with the circle is a good one. I don't know if they still teach that ... mentioned it to xyzzy the other day and he didn't know of it.

Hmmm, I should have posted my diagram. It's the first quadrant of the unit circle with radii drawn at 20, 40, 60 and 80 degrees. I think the exercise was just finding the lengths of the adjacent and opposite sides, and observing that they

really wereequal to the cosine and sine [respectively] of the angle at the origin.Do you think that would be amazing to a fourth former? It is a proof of sorts, that the sine and cosine are what we say they are. Perhaps that was what I liked about it.

Perhaps I should look up my fourth form maths teacher and ask her how she did it.

You know what I mean. The adj and opp sides of the triangles formed by dropping perpendiculars from the ends of the radii to the horizontal axis.

Plus you get a star.

mmm ... so I guess the question then is: "how did you know what the cosine and sine of that angle was supposed to be?"

If you got them from Eton tables then all it proves is that someone drew a bunch of such triangles and measured their sides and put them in the book.

IIRC, the sheer amount of work involved in such an undertaking was part of what impressed me. Sort of, "oh, so is

thatwhere they get it from?".What I want to know is how my calculator works out the trig ratios. Is it looking up a table, and if so, how does it get the values for angles which aren't in the table? Does it interpolate or approximate? And if it's not looking up a table - surely a calculator isn't smart enough to sum the infinite series form of sine and cosine?

Your calculator might well sum the Taylor series, but after reducing the range of the argument to less than 360 degrees, and maybe even just one quadrant or even 45 degree thingy. It's not like it's hard .. very simple little loop to write, and there are only a few functions it needs to do it for. sin, cos, ln and exp are all trivial and all converge rapidly.

My HP28S calculator, which says (c)1986 on the back, can give you a Taylor series for any (differentiable) function, and do it symbolically.

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