This post shows how to use Geogebra to demonstrate this fundamental truth in geometry and hopefully demystify Trigonometry to a certain extent.
As with all things Geogebra, I always try to start with a blank sheet (see other posts on this here and here). This time, I’m not using the Geogebra app itself but just launching it from within a Chrome browser window which works pretty well.
Once it is launched, I right click in the middle to remove the axes, but I am going to leave the grid on.
Then I create the triangle by constructing a line, a perpendicular line…
…and a third point which I then join to create a triangle using the polygon tool.
Next, measure the base angle of the triangle remembering the convention that angles are measured in an anti-clockwise direction.
The next bit is a tad fiddly. You need to right click on the line segment to change the label to “value”. Then do the same for the other two sides of the triangle so that you now have one angle and all three side labelled.
So far, this has taken about 2 minutes to create from a blank screen. You could do it in advance of the lesson, but I think it is worth doing it in front of the class, maybe having practiced it a couple of times. Using “something I created earlier” is less powerful – it looks like some sort of trick, somehow.
Now you have everything set up you can start moving the points as shown here.
I start by moving point B, thus keeping the angle fixed. I would ensure students have calculators in front of them and ask them to calculate opposite divided by adjacent. Then move the triangle to get different values for side lengths. Then do the calculation again. The answer is the same, of course. I might ask them how they could get that directly from the angle (tan angle). Depending on where the discussion goes with that, I might then move on to look at sin and cos.
Finally, I always like to talk about how things were done in the old days, being careful to point out that I’m not that old and that I didn’t actually use these…
I explain that the sin button on your calculator is basically just looking up the values in the sin column of a table like this – not actually true, I know, but it helps understand what’s going on so that’s OK for me!