6 ms·

Touch Mathematics – Trigonometry

117 points by kjhughes 13y ago

prezjordan 13y ago
Ha, cool visualization of sec, csc, and cot! Never really knew how they fit in the unit circle.
rupert_murdaaa 13y ago
/* DON'T EDIT BELOW THIS LINE */ OK
TempleOSV2 13y ago
I have a master's in rocket science. Control Systems. http://www.youtube.com/watch?v=WBCZBOB3LCA http://www.youtube.com/watch?v=WBCZBOB3LCA http://www.templeos.org/files/ASU_Transcripts.pdf http://www.templeos.org/files/ASU_Transcripts.pdf
Hoozt 13y ago
This makes me feel like an idiot.
huherto 13y ago
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.
- Rzor 13y ago
  >Idea: Can you make a list of questions, exercises, or problems can be answered by interacting with the model. This would be great.
- seivan 13y ago
  Perfect idea! Now, for someone to code/write it as well.
saraid216 13y ago
Neat as a reference chart. For amusement, add another dozen trigonometric functions?
kalid 13y ago
Shameless plug, I just wrote about how to learn trig without onerous memorization (http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/). A reader made some simulations based on the dome/wall/ceiling metaphor (https://www.desmos.com/calculator/az45nwnmis 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.
- saraid216 13y ago
  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.
  - gfodor 13y ago
    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.
    - circleguy 13y ago
      What's a circle if not all the right triangles (up to a choice of units)?
      - 13y ago
matthewtoast 13y ago
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.
- jasode 13y ago
  Very nice work. Bret Victor also has demos showing "dynamic" manipulation of math. http://worrydream.com/#!/KillMath 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://en.wikipedia.org/wiki/User:LucasVB/Gallery
- vlasev 13y ago
  This looks decent but I think it could be made a lot better. Right now it's kind of messy with all the functions and all the lines and numbers on the circle there at the same time. It would be better to be able to select those functions you wish to view.
  - vlasev 13y ago
    Reason for disagreement?
- ht_th 13y ago
  > 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.
saganus 13y ago
This is amazing. I never understood trig to this level. Seeing it geometrically makes it so much clearer. Very nice.
tantalor 13y ago
Please add support for touch events! http://www.html5rocks.com/en/mobile/touch/ http://www.html5rocks.com/en/mobile/touch/ Because "touch".
e3pi 13y ago
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.
crncosta 13y ago
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.
keithpeter 13y ago
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

117 points by kjhughes 13y ago

prezjordan 13y ago

Ha, cool visualization of sec, csc, and cot! Never really knew how they fit in the unit circle.

rupert_murdaaa 13y ago

/* DON'T EDIT BELOW THIS LINE */ OK

TempleOSV2 13y ago

I have a master's in rocket science. Control Systems. http://www.youtube.com/watch?v=WBCZBOB3LCA http://www.youtube.com/watch?v=WBCZBOB3LCA http://www.templeos.org/files/ASU_Transcripts.pdf http://www.templeos.org/files/ASU_Transcripts.pdf

Hoozt 13y ago

This makes me feel like an idiot.

huherto 13y ago

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.

Rzor 13y ago
>Idea: Can you make a list of questions, exercises, or problems can be answered by interacting with the model. This would be great.
seivan 13y ago
Perfect idea! Now, for someone to code/write it as well.

saraid216 13y ago

Neat as a reference chart. For amusement, add another dozen trigonometric functions?

kalid 13y ago

Shameless plug, I just wrote about how to learn trig without onerous memorization (http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/). A reader made some simulations based on the dome/wall/ceiling metaphor (https://www.desmos.com/calculator/az45nwnmis 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.

saraid216 13y ago
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.
- gfodor 13y ago
  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.
  - circleguy 13y ago
    What's a circle if not all the right triangles (up to a choice of units)?
    - 13y ago

matthewtoast 13y ago

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.

jasode 13y ago
Very nice work. Bret Victor also has demos showing "dynamic" manipulation of math. http://worrydream.com/#!/KillMath 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://en.wikipedia.org/wiki/User:LucasVB/Gallery
vlasev 13y ago
This looks decent but I think it could be made a lot better. Right now it's kind of messy with all the functions and all the lines and numbers on the circle there at the same time. It would be better to be able to select those functions you wish to view.
- vlasev 13y ago
  Reason for disagreement?
ht_th 13y ago
> 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.

saganus 13y ago

This is amazing. I never understood trig to this level. Seeing it geometrically makes it so much clearer. Very nice.

tantalor 13y ago

Please add support for touch events! http://www.html5rocks.com/en/mobile/touch/ http://www.html5rocks.com/en/mobile/touch/ Because "touch".

e3pi 13y ago

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.

crncosta 13y ago

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.

keithpeter 13y ago

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