Wednesday, September 29, 2010
Wednesday Math, Vol. 129: The trouble with trigonometry
Here's the thing about trig. With a tiny amount of information, you can figure out a whole passel of stuff. Let's assume that this is the unit circle, which means it all the points where
x² + y² = 1. The x coordinate is the cosine of the angle and the y coordinate is the sine. There are four other trig functions, but once you've got sine and cosine, they can be derived as follows.
tangent = sine/cosine
cotangent = cosine/sine
secant = 1/cosine
cosecant = 1/sine.
If I give you anyone of these values and I tell you what quadrant the angle is in, you can find all the rest of the values. That means two pieces of information gives you five more.
Except, not really. Two pieces of information gives you forty seven more. Here's why.
I've marked eight angles in the picture, a and seven more that have some symmetrical relation to a. If we say the values at angle a are (x, y), then we also know the values at the other seven points.
a °-> (x, y)
(90 - a)°-> (y, x)
(90 + a)°-> (-y, x)
(180 - a)°-> (-x, y)
(180 + a)°-> (-x, -y)
(270 - a)°-> (-y, -x)
(270 + a)°-> (y, -x)
(360 - a)°-> (x, -y)
Confused yet? So are my students. It's a lot of stuff to grasp, but it's that first glimpse into symmetry, the most important concept in mathematics. Just a little information and the assumption of symmetry can unlock so many tricky puzzles if your mind works that way.
I am completely convinced not every mind works that way, and to have a mind that works that way and thinks this is kind of cool... an even smaller set.
The struggle continues.