Chapter 29
Differentiation of parametric
equations
29.1 Introduction to parametric
equations
Certain mathematical functions can be expressed more
simply by expressing, say, x and y separately in terms
of a third variable. For example, y =r sin θ, x =r cos θ.
Then, any value given to θ will produce a pair of values
for x and y, which may be plotted to provide a curve of
y = f (x).
The third variable, θ, is called a parameter and the
two expressions for y and x are called parametric
equations.
The above example of y =r sin θ and x =r cos θ are
the parametric equations for a circle. The equation of
any point on a circle, centre at the origin and of radius
r is given by: x 2 + y 2 =r 2 , as shown in Chapter 13.
To show that y =r sin θ and x =r cos θ are suitable
parametric equations for such a circle:
Left hand side of equation
= x
2
+ y
2
= (r cos θ)
2
+ (r sin θ)
2
= r
2 cos
2
θ + r
2 sin
2
θ
= r
2
cos
2
θ + sin
2
θ
= r
2
= right hand side
(since cos
2
θ + sin
2
θ = 1, as shown in
Chapter 15)
29.2 Some common parametric
equations
The following are some of the most common parametric
equations, and Fig. 29.1 shows typical shapes of these
curves.
(a) Ellipse
x = a cos θ, y = b sin θ
(b) Parabola
x = a t 2 , y = 2a t
(c) Hyperbola
x = a sec θ, y = b tan θ
(d) Rectangular x = ct, y =
c
t
hyperbola
(e) Cardioid
x = a (2 cosθ − cos 2θ),
y = a (2 sinθ − sin 2θ )
(f ) Astroid
x = a cos 3 θ, y = a sin 3 θ
(g) Cycloid
x = a (θ − sin θ ), y = a (1− cos θ)
29.3 Differentiation in parameters
When x and y are given in terms of a parameter, say θ,
then by the function of a function rule of differentiation
(from Chapter 27):
d y
dx
=
d y
dθ
×
dθ
dx