Eliminating the Parameter of Parametric Equations (Cambridge (CIE) AS Maths: Pure 2): Revision Note

Eliminating the parameter of parametric equations

What does eliminating the parameter mean? 

• In parametric equations, x = f(t) and y = g(t)

• There is still a connection directly linking x and y

  • This will be the Cartesian equation of the graph

How do I find the Cartesian equation from parametric equations? 

• STEP 1
  Rearrange one of the equations to make t the subject

  • Either t = p(x) or t = q(y)

• STEP 2
  Substitute into the other equation

• STEP 3
  Rearrange into the desired (Cartesian) form

How can I eliminate the parameter using trigonometric identities?

• STEP 1
  Rearrange both equations into the forms “cos t = …” and “sin t = …”

• STEP 2
  Square both sides of both equations

• STEP 3
  Add the equations together

• STEP 4
  The trig identity “sin² x + cos² x ≡ 1” eliminates t

• STEP 5
  Rearrange into desired (Cartesian) form

  • This technique is seen in Trigonometric Identities