Chapter 29

Differentiation of parametric

equations

29.1 Introduction to parametric equations

Certain mathematical functions can be expressed more
simply by expressing, say,xandyseparately in terms
of a thirdvariable. For example,y=rsinθ,x=rcosθ.
Then, any value given toθwill produce a pair of values
forxandy, which may be plotted to provide a curve of
y=f(x).
The third variable,θ, is called aparameterand the
two expressions for yandx are called parametric
equations.
The above example ofy=rsinθandx=rcosθare
the parametric equations for a circle. The equation of
any point on a circle, centre at the origin and of radius
ris given by:x^2 +y^2 =r^2 , as shown in Chapter 13.
To show thaty=rsinθandx=rcosθare suitable
parametric equations for such a circle:

Left hand side of equation

=x^2 +y^2

=(rcosθ)^2 +(rsinθ)^2

=r^2 cos^2 θ+r^2 sin^2 θ

=r^2

(

cos^2 θ+sin^2 θ

)

=r^2 =right handside

(since cos^2 θ+sin^2 θ= 1 ,as shown in

Chapter15)

29.2 Some common parametric equations

The followingare some of the most common parametric

equations, and Fig. 29.1 shows typical shapes of these

curves.

(a) Ellipse x=acosθ,y=bsinθ

(b) Parabola x=at^2 ,y= 2 at

(c) Hyperbola x=asecθ,y=btanθ

(d) Rectangular x=ct,y=

c

t

hyperbola

(e) Cardioid x=a(2cosθ−cos2θ),

y=a(2sinθ−sin2θ)

(f ) Astroid x=acos^3 θ,y=asin^3 θ

(g) Cycloid x=a(θ−sinθ),y=a( 1 −cosθ)

29.3 Differentiation in parameters

Whenxandyare given in terms of a parameter, sayθ,

then by the function of a function rule of differentiation

(from Chapter 27):

dy

dx

=

dy

dθ

×

dθ

dx