This is the story about the * sinus* and * cosinus* brotherhood. Sinus and Cosinus
(Sin() and Cos() for friends) are bro's. They live in a neighbourhood called
Unity Circle.

Cos and Sin are both Values. While Cos represents the 'projected length' of the Unity Radius on the horizontal axis, Sin represents the 'projected length' of that very same Radius on the vertical axis. This is not always the case, but basically you can say Cos is the projected length on the adjacent side (adjacent to the angle) and Sin is the projected length on the side across from the angle.

When that angle (*alpha*, i.e. the Greek 'a')
is zero, Cos is consequently 1 and Sin is 0. When the angle is 90 degrees, Cos
is 0 and Sin is 1. And you can see what the values of Cos and Sin are for other
angle values above. You've probably seen already that at 180 degrees, Cos is -1
and Sin is 0, while Cos is 0 and Sin is -1 at 270 degrees. And *l'histoire ce
repête* after 360 degrees. All right, so Cos and Sin, they're both
borderline weirdo's as only those scientific Math types can handle *negative*
lengths. Totally ridiculous in 'real life out there', yet *negative* length
does exist in their neighbourhood. And it is thriving.

The Unity Circle is a circle. But it is a bit of a special circle, and
feeling blue today. It has a radius of 1 (and thus a diameter of 2). Again, the
Math propellerheads have trouble with units as it ain't 1 *inch* nor 1 *millimetre*
let alone 1 *mile*. It's just '*one*'. Just '*one*'. 1. Now we
see our Radius (which has length '1' ) rotating. If a line, called 'radius',
rotates around a point, it's other 'end point' (the blue head) draws a circle.
The Unity Circle in *this* case.

The length of the radius is always 1. At *all* times. But sometimes the
other geeks like to know what the 'projected length' is: the part of the length
across *x* or *y* or other axis. For that they have a collection of
smart folks doing all sort of *goniometric stuff*. And these people use Cos
and Sin a *lot*. Look at the demo above and see the red Cos line and Green
Sin line grow and shrink in length while Radius rotates.

Meanwhile there's the red dotted Cos() and green dotted Sin() curve being
built next to the Unity Circle. The horizontal axis of this graph depicts *alpha*
(the angle of rotation, going from 0 to 720 degrees in this demo), while the
'projected lengths', i.e. Cos() and Sin() values for each rotation value are
displayed vertically, going from +1 to -1. Old values fade away ('**Trail
distance**' and '**Trail size**' allow you to change this), while the scene
repeats itself every 720 degrees: see the blue dotted angle of rotation grow to
720 degrees (i.e. going around *twice*) and restart at 0 degrees.

This Flash demo was created as part of a * basic* basic math course, using Math
and Flash. See the downloadable FLA for the code, comments and 'special
activity' you need to perform when converting goniometric math formulas to
Flash. For one, the most important difference between computers and math is that
math often uses a orthogonal coordinate system (x axis and y axis are perpendicular
to each other; both have the same size of a length '1' while
increasing positive values on the x axis progress in the right (positive
horizontal) direction and increasing positive values on the y axis progress to
the top (positive vertical) direction.

With computers, on the other hand, increasing positive values on the y axis
progress to the *bottom* (*negative* vertical) direction. This
requires some additional +/- sign swapping before formulas work with computer
displays (and Flash). Sigh.

Note the multiple application of the 'Circular Preloader' mechanism for showing the growing (blue dotted) angle of rotation.

[*Hm, this needs some more graphics and a better explanation if this is going
to be a stand-alone gonio 'lesson' for Flash. :-( * ]

You might want to have a look at: http://www.math.com/school/subject2/lessons/S2U4L1GL.html, http://www.math.com/school/subject2/lessons/S2U4L2GL.html and http://www.math.com/tables/algebra/functions/trig/overview.htm. And maybe some other stuff at http://www.math.com/... (Try plotting 'y=sin(x)' for x = 0 to 7 at http://www.math.com/students/solvers/graphs/plot.htm, for instance. ;-)