Friday, August 24, 2007

A Trig Mnemonic Revisited with Texify!

Some time ago, I posted a piece about math mnemonics. Buried near the bottom was my feeble attempt to make a table showing a well-known (?) fascinating pattern for sin and cos values for the common angles in Quadrant I. Over the years, some students have found this to be as useful as memorizing ordered pairs on the unit circle or deriving everything from 30-60-90 and 45-45-90 (which I still prefer personally). I've seen students make this table at the top of their trig unit exam - they figured it was worth the effort! I'm reprinting this today using an image created in LaTeX and the absolutely wonderful and easy to use Texify website. This has been a real boon for those using Blogger since LaTeX has not yet been supported. Many math bloggers have been using it for a while now and I'm sure they appreciate its power and simplicity as much as I do. Its author is Andrey Burkov and Ars Mathematica gives him proper credit here. Certainly if an old dog like me can learn new tricks like this, anyone can! By the way at the Texify site, there is an extremely well-written tutorial with many examples to follow. I suspect I will be using this a lot for my new posts and perhaps cleaning up my old. Let me know if the table below is as readable as it appears to me. and, of course, if you like the pattern, you can tell me that too!

The original post used the klutziest of notations and was barely readable. This should be a lot better! I omitted the row for the tan function which is just the quotient of rows 2 and 3:


\begin{matrix}&&0^\circ&&30^\circ&&45^\circ&&60^\circ&&90^\circ\\\sin&&\frac{\sqrt0}2&&\frac{\sqrt1}2&&\frac{\sqrt2}2&&\frac{\sqrt3}2&&\frac{\sqrt4}2\\\cos&&\frac{\sqrt4}2&&\frac{\sqrt3}2&&\frac{\sqrt2}2&&\frac{\sqrt1}2&&\frac{\sqrt0}2\end{matrix}

4 comments:

Jonathan said...

Nice layout (though a bit large)

Sine squared plus cosine squared equals one jumps right out...

Sqr(1)/3 and Sqr(8)/3
Sqr(2)/3 and Sqr(7)/3
...

Why do these follow the same pattern, but not produce "nice" angles?

Dave Marain said...

Thanks, Jonathan--
I intentionally made the table large but, as you know, there are simple LaTeX commands to reduce the font size. I'm still learning the nuances of the syntax but it seems fairly straightforward. I can't compete with Wordpress folks who have had LaTeX support for months but I'm having fun with it. Texify is an awesome tool and the developer should be lauded for it.

As far as the 'coincidence' of this pattern or whether there are other similar patterns, I suspect there might be more to this. For example, the special angles in the table are multiples of 15 degrees. The sin and cos of 15 involves both the square root of 3 and the square root of 2 (using difference or half angle trig identities). Another set of angles with fascinating properties are the multiples of 18 degrees: 18,36,54, and 72. The cos(36) is a fairly interesting number, half the golden ratio! The sin and cos of the other angles I listed are related to that ratio. One could find these relationships in the golden triangle and the regular pentagon. I'm thinking about an investigation involving both that triangle and the pentagon but that will take some work and lots of graphics (not my strength). There are many excellent websites that give the facts but you know me. I need to write an activity for students that develops the key ideas in stages, posing many open-ended questions of increasing difficulty. Anyone interested? Actually, I think I could use some collaboration here...

Jejomar said...

The picture at the bottom just saved me.

I'm stuck in the middle of memorizing a table of trigonometric values and... it was heaven sent! THANK YOU!

watchmath said...

Its been almost two years and it seem you haven't solve your latex problem.
I believe I have solved the latex problem for Blogger user. You can check my new article about installing latex on blogger: http://watchmath.com/vlog/?p=438