Touch Mathematics – Trigonometry(touchmathematics.org)
touchmathematics.org
Touch Mathematics – Trigonometry
http://www.touchmathematics.org/topics/trigonometry
13 comments
It's not really clear what "learning trig" actually means.
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.
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.
I don't think you can get away describing trig in a sentence without using the word 'circle'. Really I'd argue trig is at its core more about the circle.
But it's trigonometry, not anometry.
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.
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.
What's a circle if not all the right triangles (up to a choice of units)?
The solution set to equations of the form X^2 + Y^2 = R^2?
The boundary of a ball in a 2-dimensional, euclidean metric space?
An equivalence class of Gaussian numbers based on their norms?
The boundary of a ball in a 2-dimensional, euclidean metric space?
An equivalence class of Gaussian numbers based on their norms?
Look at the beauty of the sin function and tell me that has more to do with triangles than the circle. sin and cos are the length of the projected vector along each axis as a point moves about the circle. connecting those points forms a triangle, sure, but it seems less fundamental.
I'd absolutely concede that the sine function is more about a circle than it is about a triangle. If you could really say that the sine function is about a geometry at all.
Thanks for the feedback! I'm in general agreement with your second summary; basically, the properties of triangles end up appearing all over math, so our trig terminology/relationships become generally applicable, especially to circles and other repeating patterns.
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!
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!
I like pointing out to students of classical mechanics that the first year is all about pointing out ramifications of one assumption, one mindlessly simple, underwhelming equation: constant acceleration. If you know a bit of calculus. It goes from being a terrifyingly large amount of memorization and calculation to... "Oh."
Author. Looking back at this project now, I think the real takeaway is that the act of programming can teach. (Even when you program like I do.) I built this to learn concepts that had always frustrated me in school: fundamental math that I was fed up, as a 27 year old, never having understood. Building this really helped me wrap my mind around trigonometry for the first time, and I still open up this visualization when I need a reminder.
> I think the real takeaway is that the act of programming can teach.
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.
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.
Very nice work. Bret Victor also has demos showing "dynamic" manipulation of math.
http://worrydream.com/#!/KillMath
There's also the Lucas gallery but they are straight animations and not manipulated by user.
http://en.wikipedia.org/wiki/User:LucasVB/Gallery
http://worrydream.com/#!/KillMath
There's also the Lucas gallery but they are straight animations and not manipulated by user.
http://en.wikipedia.org/wiki/User:LucasVB/Gallery
Ha, cool visualization of sec, csc, and cot! Never really knew how they fit in the unit circle.
Please add support for touch events!
http://www.html5rocks.com/en/mobile/touch/
Because "touch".
http://www.html5rocks.com/en/mobile/touch/
Because "touch".
Nice work
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
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
A nice simple identity I obtained in high school learning trigonometry:
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.
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.
Pretty cool graphics. But I don't know what to do with or take advantage of it. How do I validate that I am actually understanding what I am suppose to understand?
Idea: Can you make a list of questions, exercises, or problems can be answered by interacting with the model.
Idea: Can you make a list of questions, exercises, or problems can be answered by interacting with the model.
This is really a beautiful work, congratulations!
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.
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.
This is amazing.
I never understood trig to this level. Seeing it geometrically makes it so much clearer. Very nice.
I never understood trig to this level. Seeing it geometrically makes it so much clearer. Very nice.
This makes me feel like an idiot.
Neat as a reference chart. For amusement, add another dozen trigonometric functions?
/* DON'T EDIT BELOW THIS LINE */
OK
OK
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.