I have students use geometry software to construct a right triangle with a side of x between the 90 degree angle and angle B. Now make B = 10 and compute the ratios opp/hyp, adj/hyp, opp/adj and put the numbers in a table. Now repeat for B = 20, 30, 40, 45, 50, 60, 70, 80.
Now change the length of the included side from x to x+3, or some other number. Repeat the measurements.
I can assure variety by letting x = # of letters in first name. Now we compare table values. Why are the ratios for B = 10 the same for all students and different values of x? Students quickly catch on the the fact that given a right triangle with a certain angle, the ratios will be the same. They also notice that sin(x) = cos(90-x).
With Geometric Supposers and GeoExplorer, students constructed a sequence of triangles. With Geometry Inventor and Geometer's Sketchpad (and also Cabri, I believe), students construct one triangle and merely drag one vertex to change the values. With the latter, the table can be recorded right on the screen.
An extension could be to investigate ratios in similar triangles that are not right triangles. A triangle could be constructed and then dilated to construct a similar triangle. But what would this prove? By definition, in similar triangles, right or otherwise, corresponding sides are in the same ratio. The concept of sine is attached to an angle, not to a particular triangle. In a right triangle if you know one acute angle, you know them all. This is not the case in a non-right triangle. Is this the message? Eileen Schoaff Bflo State College