Question

Why are parametric equations used instead of putting it all into one cartesian equation?

For example, it would seem simpler to write `y=cos((x+4)/3)` instead of `x=3t-4,y=cost`

Answer 1

Another good example could be in Mechanics where the horizontal and vertical position of an object are dependant upon time, so we can describe the position in space as a coordinate:

`P=P( \ x(t), y(t) \ )`

Another reason is that we always have an explicit relationship, for example the parametric equations:

`{ (x=sint), (y=cost) :}`

represents a circle with a 1-1 mapping from `t` to `(x,y)`, whereas with the equivalent cartesian equation we have the ambiguity of sign

`x^2 + y^2=1`

So for any `x`-value we have a multi-valued relationship:

`y = +-sqrt(1-x^2)`