Sine and Cosine Made Easy

Published on Sunday, April 03, 2011 in , , , , ,

Jim Belk's unit circle illustrationNow that I've helped make radian and degree conversion easier in my previous post, it's time to tackle the even more fearful part of the unit circle, the sine and cosine aspect!

Instead of working out sine and cosine in your head, though, you're going to have to work them out on your fingers.

On the last page of my unit circle tutorial, I include the following video, which teaches the basic finger trick we'll be discussing in this post:

We're not going to worry about tangents and cotangents in this post. However, note that this video is referred to as part 1, and the teacher mentions that she'll cover the other quadrants in another video. Unfortunately, as near as I can tell, that second video was never posted.

That's OK, because inspired by this video, I developed my own approach and mnemonics for covering the other quadrants. Make sure you've got the basics down as taught in the video above before dealing with the other quadrants.

The mnemonic I developed to help deal with this approach involves imagining that you live upstairs from a mom-and-pop store, where you work during the day for your parents (Mom and Pop). Before I explain it, here's the mnemonic itself, starting with Pop waking you up for the day:

  • First, Pop comes upstairs to get you up.
  • Second, He asks you to mop down the floor in the living area.
  • Third, Mom comes upstairs to update you with some news.
  • Fourth, You're taken downstairs to meet the new employee, whose name is Pam.
Simple enough? It'll be even simpler to remember when I explain the significance of each step.

The video above teaches the hand trick for quadrant 1, which is in the upper right, so let's start there. The hand trick is simple there, as both the cos and sin are positive, and the fingers start palm towards you, with the left pinky being 0° up to the thumb being 90°. The important factors here are that the cos and sin are both positive numbers, and that you count in an upward direction on your hand.

How does this relate to the mnemonic? Remember who comes to get you up out of bed? It's Pop. PoP = Plus, Plus! In which direction is Pop heading? He's heading UP to get you UP and out of bed. This will help you remember to go UPward on your fingers.

Let's try a quick problem with this knowledge before we move on, such as the cos and sin of 60°. We're working in quadrant 1, so we recall that PoP goes UP first. The fingers go in upward order 0° (pinky), 30° (ring finger), 45° (middle finger), 60° (index finger) - Ah! That's the one we're looking for! We fold that in, and we see 1 finger above (giving up part of the cos), and 3 fingers below (giving us part of the sin). That translates into:

A quick adjustment for the square root of 1, which is 1, is made. Since PoP helps us remember that both signs are positive, we're done here.

Let's move onto quadrant 2, which is the upper left one. Remember, when dealing with circles, you count the quadrants in the order as if you're making a C (for circle!). We're in quadrant 2, so what happens second in our story? You're handed a MoP and told to MoP DOWN the floor.

This means that you're going to be working downard on your hand, from the thumb to the pinky. The thumb will still be 90°, but now the fingers will represent the angles as follows: 120° = index finger, 135° = middle finger, 150° = ring finger, and 180° = pinky. MoP, of course, reminds us that the signs will now be Minus, Plus.

With this knowledge, what are the cos and sin (respectively) of 135°? We head down from the thumb to the middle finger, since the middle finger represents 135135°. We fold that in, noting that there are 2 fingers above (helping us with the cos), and 2 fingers below (helping us with the sin). This would seem to suggest that the answer is:

However, this neglects to take the signs into consideration. Remember the MoP? Since that means Minus, Plus, we adjust our answer appropriately:

Got the idea? We're already half done! Remember, through all these quadrants, you're always working with the left hand, and with the palm facing you.

We're moving to quadrant 3, in the lower left. What happens in the 3rd part of the mnemonic? MoM comes UPstairs with an UPdate. You're already ahead of me, aren't you?

The pinky, of course, is still 180°, and we go UP (naturally) from there: 210° = ring finger, 225° = middle finger, 240° = index finger, 270° = thumb.

Think you can handle the cos and sin of 210°? Try it on your own before reading further. Don't forget to get help from MoM!

What did you come up with? We bend in the ring finger, the 210° finger. That 3 fingers over, and 1 below. This, along with the signs hinted by MoM, gives us:

To round this out with quadrant 4 (correct - the lower right one), we'll figure out 315°. What happend in the 4th and final part of the mnemonic? You go DOWNstairs to meet PaM, the new employee. Since you probably have the idea by know, which fingers go with which angles?

Going DOWN from the 270° thumb to the 360° pinky, we have a 300° index finger, a 315° middle finger, and a 330° ring finger.

How about that 315° cos and sin, not forgetting our new employee PaM? Folding the middle finger in, we have 2 above and 2 below. Which gives us what for an answer?

PaM is Plus, Minus, right? That's why the signs are placed where they are.

Now that you've got the idea of how to use the mnemonic, here it is again with the important parts in bold:
  • First, PoP comes UPstairs to get you UP.
  • Second, He asks you to MoP DOWN the floor in the living area.
  • Third, MoM comes UPstairs to UPdate you with some news.
  • Fourth, You're taken DOWNstairs to meet the new employee, whose name is PaM.
Yet again, missing information provides inspiration for a little creativity. Thanks go to that one missing video.

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