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 (or, in degrees, ) 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 or .
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 , starting with my palm facing in, I put down the second finger from the left and flip over my hand:
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 .
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.