Some relations are better written in terms of a third variable instead of x and y. These equations are called parametric equations and t is often used as the third variable. t is called a parameter.

Parametric equations are useful when dealing with the conic sections and other relations in geometry.


The circle x2 + y2 = 9 can be represented by the parametric equations

x = 3 cos t
y = 3 sin t

This can be shown by substituting into x2 + y2 = 9

x2 + y2 = (3 cos t)2 + (3 sin t)2  
  = 9 cos t + 9 sin t  
  = 9( cos t + sin t) (as sin x + cos x = 1)




Differentiation of Parametric Equations

Differentiation of parametrically defined relations is best done using the Chain rule. It is easiest to follow using Y12_Parametric_Equations_01.gif notation.


Using the equation x2 + y2 = 9 which has parametric equations x = 3 cos t and y = 3 sin t.

Differentiate each parametric equation:

Y12_Parametric_Equations_02.gif = -3 sin t

Y12_Parametric_Equations_03.gif = 3 cos t

Using the Chain rule Y12_Parametric_Equations_04.gif             ør  (Y12_Parametric_Equations_05.gif )