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Tuesday, February 1, 2011

Files for Riley's Intro to Trig

I assigned this to be done outside class today, since tomorrow's lesson is "Trig on the Unit Circle." You should go read it because otherwise I'd just be copying and pasting the whole thing.

I'm a little flail-y, still, about where to take it tomorrow. I have never had success getting the cherubs to see the connection between the coordinates on the unit circle and sine and cosine. Or when I have, it's been fleeting. I was thinking we'd discuss the patterns and shortcuts, and then I'd pose "Let's come up with a way to find precise coordinates for 57.4 degrees without having to draw it." But I'd love to hear better ideas.

Anyway, I thought it would be helpful to make the files I'm using available. They are here. One's a GSP and the other is an Excel.

1. You could reframe "Let's come up with a way to find precise coordinates for 57.4 degrees without having to draw it." as a problem of converting from polar coordinates to cartesian coordinates. You could then come up with plenty of real-world applications (e.g. sonar, radar, orienteering)

2. Introducing polar coordinates is a little too beyond-the-scope. It's a great idea but this course is so pressed for time.

3. Maybe go to the graphs of sine and cosine by plotting points? Right now you've got three pieces of data: angle, x-coord, and y-coord. You also have a plot of x vs. y (the circle). What if you did the other pairs: angle vs. x or angle vs. y. Fill in the gaps with what you think the graph should look like?

4. I should have been more clear about the goal. I want them to come to the understanding that the coordinates of any point on the unit circle are (cosQ, sinQ) where Q is the central angle. And that's all for now - I don't want to teach a whole other unit tomorrow.

The graphs of (Q, cosQ) and (Q, sinQ) are covered extensively in a following unit. I'd rather not get into it now.

5. All suggestions of course - feel free to steal or ignore.

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Challenge the class to figure out sin57 with just a ruler and protractor - no calculator. Hopefully some bright spark will put the equipment together with last night's homework, and draw a right-angled triangle with a diagonal of 1.

So, same challenge - cos23. This time, draw the triangle yourself, putting the 'angle' in the same place.

Finally, draw the unit circle in. Pick a third point on the circle, in the same quadrant. Draw an 'x' there.

Hopefully someone (perhaps someone who's done the homework) will now tell you that you should draw the triangle, and that the two straight sides will give you sin and cos of the angle.

6. By the way - I still remember a lesson when I was twelve, and had just been introduced to right-angled trig.

Our teacher told us there was a way to calculate sine and cosine without using a calculator, and simply told us the trick. Draw a unit circle; mark your angle; and measure the co-ordinates.

We got in most of a lesson's practice, and it helped us remember that sin used the opposite and cos used the adjacent. Years later, when I studied complex numbers... well, it was all a lot easier.

If your school includes those age groups too, you might want to talk to some of those teachers about whether they could include something like that.

7. How about some triangles cut out to the same dimensions of the circles you gave out? Use ruler/protractor to find measures of sides/angles. Find sine/cosine for those triangles with those measurements. Slide triangles onto their homework and see if you can match any of the numbers.

8. Are your students familiar with SOH CAH TOA and right triangles?

This is how I introduced the unit circle: I started with a circle of radius 2 and drew a triangle with a 30 degree center angle. Because this is one of the special right triangles, students are able to see that the x-coordinate is sqrt(3) and the y-coordinate is 1. We do the same thing with a 60 degree triangle. Then we drew a 45 degree triangle, but this time on a circle with radius sqrt(2). Then we asked: what if we wanted to "standardize" the circle so that all the these angles were on a circle with the same radius? What if we made that radius one? By dividing all the side lengths by the radius, we get our values for sin and cos.

This might not make any sense, but it seemed to help my students make a connection!

9. SIMMS, published now by Kendall Hunt (Level 4)has a module called "Can-It" It takes some time to do, (not a one day thing) but it does the best job of teaching students about radians and how the graphs of sine and cosine are related to the unit circle. The next time I teach Algebra 2, I'm going to include it.

10. Dan, That stuff costs quite a bit. Can you tell us a bit more, so we can make our own activities that are similar?

11. Here's a video I made with GSP illustrating where the sine and cosine curves come from: