December 9, 2010

All Hail Ailles

Filed under: High Effort/Low Payoff Ideas — Adam Glesser @ 5:28 pm
Tags: ,

This post is a sequel of sorts to my tricks of the trade post describing how to quickly compute the standard values of sine and cosine in the first quadrant. This is a very nice little exercise, known by quite a few high school teachers, but unknown to everyone I know personally. This lesson assumes that the student is aware of the basic facts from geometry about 30-60-90 and 45-45-90 right triangles as well as the Pythagorean theorem. If they know the definition of sine and cosine, you can go a little further at the end—this is my target audience. As a good preface to this, you might start with Kate Nowak’s post on these special right triangles. We start with the following rectangle:

Give this to the students (I do it in groups) and ask them find the length of each side and the measure of each angle. Simple arithmetic will take them here:

At this point, perhaps with some nudging, they apply the Pythagorean theorem to get at least two of the missing sides of the inner triangle.

The last side might be too intimidating for your students. If so, suggest they work on the angle problem. Those right triangles on the bottom should look mighty familiar…

My students got stuck here. However, I channeled my help less personality and let them struggle a bit. A few decided to work on the missing side again, but all eventually came around to noticing the angle on the bottom had to be 90 degrees—I must have said, “I don’t know. Why do you think it is a right angle?” half a dozen times that day.

It will not surprise you how few notice that the triangle is a 45-45-90 right triangle. It will surprise you how many don’t see that they can now use the Pythagorean theorem again to get the last side.

At this point, the students should have no problem sorting out the last two angles.

At the end, you might try asking them to compute the sine or cosine of 15^{\circ} and 75^{\circ}. Normally, I teach these via the sum angle formulas, but this is a far more intuitive approach.

This rectangle was noticed by high school teacher Doug Ailles and is knows as the Ailles’ rectangle. Not too shabby.



  1. This is PRETTY FREAKING AWESOME. Thanks for sharing.

    Comment by Kate Nowak — December 21, 2010 @ 1:04 pm | Reply

    • Many thanks, Kate! Your comment reminds me that I need to go back to some of my old posts and update them with links to some of the things I’ve seen recently. For instance, the worksheet from your post on logarithms would make a perfect introduction to my post on logarithms.

      This packet I’m writing for my third-year review will never get finished if I don’t stop this, but it sure is fun.

      Merry Christmas!

      Comment by Adam Glesser — December 23, 2010 @ 5:16 pm | Reply

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