http://www.ltcconline.net/greenl/courses/107/PolarParam/parameq.htm Is the reason they make x(t) = 2cos(t) and y(t) = 2sin(t) because cos generally represent x and sin generally represent y? Is there an exception to this rule? And can someone give examples of the application of sin and cos in terms of x and y?
Changing (sin(t),cos(t)) to (cos(t),sin(t)) is just a rotation and a reflection, as seen here. Its entirely a matter of convention.
It's just the conventions of our graphing system, nothing more. The horizontal axis is traditionally labeled as the X axis, with positive values traditionally plotted to the right of the origin and negative values to the left, and the vertical axis is labeled as the Y axis, with positive values plotted above the origin and negative values below. So if you draw a right triangle with one vertex at the origin, its hypotenuse extending out from it with a length equal to unity, and its sides aligned to be exactly horizontal and vertical for convenience in measurement, then: The side that runs along the X axis (the horizontal side) will have a length equal to the cosine of the angle the hypotenuse makes with the X axis, and The side that runs along the Y axis (the vertical side) will have a length equal to the sine of that angle. There's nothing special about this layout, it's just the way we've standardized it to reduce the number of words we need to describe an exercise in geometry. You can draw a right triangle anywhere you want on the graph, with the axes tilted any way you want, and the hypotenuse of any length you want. But in order to describe that triangle to us you will need to specify more variables than the two given in the average textbook problem; for example: five of the six coordinates of its three vertices.
I'm not completely sure this is what you're asking, but here goes: the x and y coordinate system (the Cartesian system, for Rene Descartes) is just one out of a load of choices of coordinate systems that you can use. Lots of things are convenient to look at in terms of Cartesian coordinates but other things are not. For example, systems where there is a lot of circular symmetry, like a planet orbiting a star is easier to study in spherical coordinates. Of course, these are just coordinate systems so the reality of what happens should not depend on what coordinate system you choose - it just it is often easier to treat problems in one system or another. Cartesian coordinates (x,y) are conventionally related to circular coordinates \((r,\theta)\) by the following: \(x = r \cos \theta, \quad y = r \sin \theta\)
Triangles used to show trigonometric functions traditionally have the adjacent side horizontal, the opposite side vertical, the 90° angle forms where they meet, and the hypotenuse runs between the other ends of these sides. The sine being opposite/hypotenuse is, thus, reflected upon the y axis (ie, the vertical axis). The cosine being adjacent/hypotenuse is, thus, reflected upon the x axis (ie, the horizontal axis). Also, with this convention, the 0° angle lies horizontally to the right, which is the convention in engineering diagrams (and different from navigational compasses). Please Register or Log in to view the hidden image!
From the sounds of it you are using a farely complicated example to ask an elementary question and if i'm right that you are merely asking why they say x=cos(t) and y=sin(t) then the answere is that it doesnt have to, it depends in where where measure the angle from, in this case which happens to be the most common the angle is measured from the x-axis. it really is simple trigonometry and if you're having truoble with it i suggest you take a few steps back and start from the beggining, learn to walk before you try to run as it were.
