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#

1 Answer
Steve M
Nov 17, 2017

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) #

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