September 8, 2010

The Circle Is Now Complete

Filed under: Tricks of the Trade — Adam Glesser @ 3:46 pm
Tags: , ,

Tricks of the Trade

(with Professor Glesser)

I watched a very nice video on remembering the values of sine and cosine at the various standard angles 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} (or, in degrees, 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}) and their second through fourth quadrant analogues. My method is only slightly different and is likely well-known, but as most of my students have never seen it, I figure there is at least one person out there I can help. This also gives me an opportunity to throw out a useful mnemonic that naturally attaches itself to the unit circle.

Get your students fingering with each other!

The first step is for your students to be absolutely, 100% comfortable with the standard angles mentioned above. I will assume that this is done. Hold up your hand (either will work) with your palms facing either away or toward you, and starting from the leftmost finger, count 0, 30, 45, 60, 90 or 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}.

You don’t need to remember which value goes with which finger; you only need to be able to recite the list and stop at the correct finger. When you get to the desired angle, put the corresponding finger down. Now flip your hand over. We will use the number of fingers on the left of the bent finger to determine the cosine of the angle and the number of fingers on the right to compute the sine of the angle. The formula for the value is “ROOT FINGERS OVER 2”

For example, to compute the cosine and sine of \frac{\pi}{6}, starting with my palm facing in, I put down the second finger from the left and flip over my hand:

The number of fingers on the left of the bent finger gives you the cosine, the fingers on the right give you the sine. Just remember ROOT FINGERS OVER 2. Why the hand flip?

Truthfully, I never used it. However, I added it because I wanted to reinforce the fact that when you define cosine and sine via the unit circle (as I do), then the coordinates on the unit circle are of the form (\cos(\alpha), \sin(\alpha)).
To make this consistent with the hand trick, you need to flip the hand.

What was this mnemonic you mentioned?

It is actually pretty silly. The following (ridiculous) diagram should help a student remember where sine and cosine are positive and negative. They need only remember where the S goes and where the C goes. “S stands for summit, so it goes on the top” has worked for my students.

Furthermore, this diagram, when read clockwise, gives you derivatives of sine and cosine (as well as their negatives). That is not too shabby.



  1. This reminds me of another “trick”.


    You can create similar tables for the other trig functions.

    Comment by Avery — September 10, 2010 @ 3:16 pm | Reply

  2. […] Adam Glesser @ 5:28 pm Tags: Ailles' rectangle, trigonometry This post is a sequel of sorts to my tricks of the trade post describing how to quickly compute the standard values of sine and cosine in the first quadrant. This […]

    Pingback by All Hail Ailles « GL(s,R) — December 9, 2010 @ 5:29 pm | Reply

  3. You could do away with the flipping of the hand by naming your fingers in reverse order: 90, 60, 45, 30, 0. This also follows the angles on a unit circle where you start on the x-axis and move counterclockwise for bigger and bigger angles. As you name the fingers, you rotate in a counterclockwise motion! This seems more intuitive to me.

    Comment by Rhonda Russell — December 23, 2010 @ 7:59 pm | Reply

    • That is a great tip, Rhonda! I love it. I can see myself actually laying my hand on an overhead projector (if I had one) with a unit circle transparency, fingers spread on the appropriate angles. I can also see devoted students painting the angles on their nails to help them remember. Hmmm…I haven’t had a manicure in a while; I wonder if…

      Happy holidays, Rhonda.

      Comment by Adam Glesser — December 23, 2010 @ 8:15 pm | Reply

  4. OMG. This is so simple. Thank you. Can’t wait to do this this year!

    Comment by Fluxion Fred — January 2, 2011 @ 5:56 pm | Reply

    • I’m glad you like it. If you haven’t already, be sure to read Rhonda’s simplification above.

      Comment by Adam Glesser — January 6, 2011 @ 2:25 pm | Reply

  5. Fantabulous! I’ll use this for sure. I’ve already adapted it a bit. Use the left hand (so the right hand is free to write for most) and use the thumb as zero. I read the comments and someone already mentioned this. My students so often get these mixed up. This is GREAT!!! Thank you for sharing.

    Comment by Jeff — October 31, 2011 @ 4:29 pm | Reply

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