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Thursday, August 6, 2009

Trig Reference Angle Cheat Hand


Flip down the finger that corresponds to the angle whose sine and cosine you need.
The number of fingers to the left gives you the sine, and the number of fingers to the right gives you the cosine.

So if you flip down your index finger which corresponds to 30 degrees...
there is one finger to the left.


and there are three fingers to the right.


Try it for the fingers that correspond to the other reference angles. For example, if you flip down your pinky, there are four fingers to the left $sin{(90)} = \frac{\sqrt{4}}{2} = 1$ and zero fingers to the right $cos{(90)} = \frac{\sqrt{0}}{2} = 0$. It works!

It's just another way of organizing the cofunction behavior of sine and cosine to remember the values of five reference angles, but adults and kids both flip out when I show them. Kids especially feel that they "don't have to memorize" if they know this method.


  1. Cute!

    I think I'll use this in my calc classes, too (except with 0, pi/6, pi/4, pi/3, pi/2).

  2. This is too cool!!

    I have shared this with several people = best "mneumonic" I've seen for a while.

  3. nice. i more or less discovered
    a version of this quite a while ago
    but of course without your pizazz.
    i used to tell beginning trig students
    that when i was starting out
    i'd construct a right triangle
    every time i needed a trig value
    for a "standard" angle... but that
    in hindsight, i felt like i'd've been
    better off just memorizing
    *these few things*:
    the sine of 0, 30, 45, 60, and 90
    degrees 9 (or as CD sez, their
    radian equivalents... meanwhile,
    i'm laying out a chart... here's
    where the pizazz fails to come in...).
    then, of course, for cosine,
    we just reverse direction...

    it just amazed me when i realized
    that the \sqrt2 and \sqrt3
    could be extended right on into
    \sqrt0, \sqrt1, and \sqrt4...
    all with the same denominator.


  4. oh, and by the way...
    there were a bunch of folks
    back home that *called* me
    "owen by the way"...

    here is a recent
    post by sue v. (and comment
    from me) on a related topic.

  5. Kate, I love you!! This is great. I will share with my new colleagues when I get back to school. You need to come back to New Braunfels and visit Schlitterbahn!! I will take you to some fun places!

  6. Very nice! I love things like this, as my kids always favor methods that use their hands!

  7. Great. I have another mnemonic method. But yours definitely the best one!

  8. Aw, shucks, you are all too kind. I'm glad you like it & hope it helps your co-learners.

    @vlorbik i find some kids still prefer a table-style organizer.

  9. Thanks for sharing another great idea. I've found lots of inspiring posts on your blog.

  10. I love the chart - never seen the hand method. Thank you for sharing!

  11. Thanks Kate - I've never seen this before! Keep on blogging, sharing ideas like this can only help our Mathematics teaching.

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  13. I used this last year and loved it! However, "my" way was reversed. If you hold your hand pinky side down, and count up 0/30/45/60/90, sin is above the finger and cosine is below. Then when you flip the hand you have the tangent (pinky side over thumb side).

  14. A student of mine shared this with me last spring and I was very impressed too. Has anyone figured out where the pattern comes from? I had previously assumed we memorized trig values for 0/30/45/60/90 because they were easy to figure out using geometry. Is it just some big coincidence that this pattern works? Can it be generalized? It seems like there's some underlying elegance that I'm missing...

  15. No, there's really nothing elegant underneath it all. It works because there have to be some angles for which it works (and there can't be any that extend the pattern).

    The captcha today is "trationy"

  16. @unapologetic: Good points -- some angle has to have each of those values, and clearly you can't extend the pattern in the numerator.

    But... it still seems like a hell of a coincidence that the angles it works for are the ones which can be trivially deduced geometrically. And there are other ways of extending it -- changing the denominator and/or index of the root. Do any of those lead to similar patterns? (I only checked arcsin(sqrt(2)/3), but no dice.)

  17. The only thing I can come up with is the 45-45-90 and 30-60-90 shortcuts can be derived from cutting a square and an equilateral triangle in half, so that sort of explains the /2 consistency... Why do they involve precisely sqrt(2) and sqrt(3)...that's one of those things that is both obvious and mysterious and makes math fun to study. But I'm pretty sure that the next integer being 4 and that sqrt(4)/2 happens to evaluate to 1 is just a coincidence.

  18. Sweet! I have never seen this - Thanks for sharing!

  19. This is awesome!

    Good for simpletons like me :)

  20. Alex: Some angles have to be geometrically simple, I'd say, and they're simple precisely because the squares of their sines and cosines are simple.

    I mean.. it's like asking if there's some deep reason that a number is even if a collection of that many things can be grouped into two equal subsets.

  21. This was so cool I shared it my precalc class today. They loved it but at the same time felt silly because they were counting on their fingers. Great tip!

  22. This helped a lot. I owe you a drink. xD

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  24. Kate, what a great trick! It's gonna be a lifesaver. I've linked your post on my blog, though I don't have many readers yet... :)

  25. So, as I work through my RSS feed, I'm a little late to the party.

    Nice tip.

  26. Can you also show how to do the trig version? My precal teacher is..... as messed up as... the book that we have so she doesn't explain anything we really need to know. It's really helpful though!!!

  27. I'm not sure what you mean by "the trig version?" Can you explain more specifically what you're looking for?

  28. This is a great trick, though I'm thinking about modifying it somewhat. What if you rotate your picture (left hand showing) so that your pinky finger is where the positive x-axis goes (0 degrees,then the ring finger is 30 degrees, ending with the thumb sticking straight up at 90 degrees.)?

    Now when you pull down the ring finger, corresponding to 30 degrees, the Cosine root3/2 is on the left (like the x-coordinate would be) and the Sine 1/2 is on the right (like the y-coordinate would be). What do you think of that modification? Any flaws?

  29. We use the left hand, palm facing the student (thumb up). Then it forms quadrant I and you're looking at the actual 0 (pinkie), 30(ring), 45(middle), 60(pointer), and 90(thumb) degree angles.

    Then, cosine starts at the thumb and travels around clockwise and sine starts at the pinkie and travels around counter-clockwise.

  30. I like that enhancement - talk about hands on! Great ideas!

  31. I will be using this in my own Pre-Calc class in January. It's excellent! Thanks for sharing it!

    Mike Poliquin

  32. Could use your right hand facing you instead of left and then label the same as you have on thumb etc and it will work as Quad 1 and cos and sin in correct order :)


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