Touch Mathematics – Trigonometry
touchmathematics.org
touchmathematics.org
A reader made some simulations based on the dome/wall/ceiling metaphor (https://www.desmos.com/calculator/az45nwnmis).
The right metaphors are essential; I can now visualize the cosecant, where it'd be useful, and intuit why cot^2 + 1^2 = csc^2. Hope this helps someone.
Trigonometry, as a subject, can be fully understood in a sentence: "Triangles (especially triangles with a 90-degree angle) have special properties that make them interesting." This is not useful, of course, and worse, it fails to justify high schoolers spending most of a year on the subject.
Trig might instead be understood in this way: "Breaking down complex shapes into triangles makes it possible to find many of the values involved in the shape, because of the special properties that triangles have."
I think this is a fair summary of the topic. If that's good enough, it might be worth doing an article based around taking a complex collection of lines and curves (and draw it out of a photograph for making it feel applicable), assigning a length here and there, and then spending the entire article zooming in on specific sections in order to figure out the consequent lengths of everything else.
Really, the unit circle is important because it's the unit circle, not because it's the unit circle. A circle is 2π angles. A triangle is composed of the same kind of angles, which means that, because the diameter of the circle doesn't matter (because it's a unit circle), then it's less important what the lengths are than it is how the lengths relate to one another. I.e., what factor of a unit vector as compared to what other factor of a unit vector.
The boundary of a ball in a 2-dimensional, euclidean metric space?
An equivalence class of Gaussian numbers based on their norms?
One of the best parts of trig is that knowing one little fact (sin(x) = foo) reveals a tremendous number of other ones (inverse sine, get the original angle; then the cosine, tangent, secant, etc.). I'd like to explore some applications (geometric and non), appreciate the suggestion!
That looks like the learning theory of "constructionism" (with the n not the v) and the ideas of S. Papert and A.A. diSessa who did research about using logo for children to learn about arithmetic and physics. Interesting stuff, especially from a point of view of a programmer who thinks about learning.
http://worrydream.com/#!/KillMath
There's also the Lucas gallery but they are straight animations and not manipulated by user.
http://www.html5rocks.com/en/mobile/touch/
Because "touch".
As a teacher, what would help make this really useful on (say) an interactive whiteboard would be
1) On/off by the functions on the bottom left so that I can focus on (say) the sin function, then switch the cosine function on, then challenge students to say what happens to sin/cos for various angles then switch on the tan function
2) Line width control or just thicker
3) Switchable angle overlays (degrees for most, radians only for the advanced ones)
Thanks
z^(1/2) = ((z+1)/2)*sin(acos((z-1)/z+1))) for all z
...similar identities may be obtained using atan, etc. This came to me from the euclidean geometry construction of sqrt(z). I would like to have its construction image here, but the ascii art is too much. Perhaps Google would find it. When I messed with it, I noticed for large z, acos()--> 1, it's argument describes an angle going to zero and `bends' the left edge of the circle's subtended arc into a vertical line, say, as a zooming magnification. A dynamic screen as yours would be fun to watch the vertex of enclosed right-angle(largest) of the sqrt(z) half-chord, as you mouse drag this vertex around.
Idea: Can you make a list of questions, exercises, or problems can be answered by interacting with the model.
This would be great.
As someone also said here, should have more explanation about the subject Trigonometry. But I am sure I can show this to a kid and give the explanation too.
Again, very well done.
I never understood trig to this level. Seeing it geometrically makes it so much clearer. Very nice.
OK