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 2 t + 9 sin 2 t = 9( cos 2 t + sin 2 t) (as sin 2 x + cos 2 x = 1)
Differentiation of Parametric Equations
Differentiation of parametrically defined relations is best done using the Chain rule. It is easiest to follow using notation.
Using the equation x2 + y2 = 9 which has parametric equations x = 3 cos t and y = 3 sin t.
Differentiate each parametric equation:
= -3 sin t
= 3 cos t
Using the Chain rule ør ( )