Parametric equations give x and y separately in terms of a third variable, the parameter t. To find the Cartesian equation, eliminate t: rearrange one equation for t and substitute, or, when both use trig functions, use an identity such as sin²t + cos²t = 1. The range of t often means the parametric curve is only part of the Cartesian curve.
Eliminating the parameter
- t appears simply in one equation: rearrange for t and substitute into the other
- cos t and sin t: square and add, using sin²t + cos²t = 1
- sec t and tan t: square and subtract, using sec²t = 1 + tan²t
- cosec t and cot t: square and subtract, using cosec²t = 1 + cot²t
- Circle with centre (a, b) and radius r: x = a + r cos t, y = b + r sin t
What are parametric equations?
Instead of one equation linking x and y directly, a parametric curve gives x and y each in terms of a parameter, usually t. Feeding in values of t traces the curve point by point. The Cartesian equation is the direct link between x and y, found by getting rid of t.
Eliminating t by substitution
When t appears simply in one of the equations, usually to the first power, rearrange that one for t and substitute into the other. Choose the easier equation to rearrange. Check your result by picking a value of t and confirming the point satisfies your Cartesian equation.
Eliminating t with a trig identity
When x and y both involve trig functions, you cannot rearrange for t easily. Make each trig function the subject, then use the identity that links them: square and add for sin and cos, or square and subtract for sec and tan, or for cosec and cot.
If the coefficients of cos t and sin t are equal, the curve is a circle. If they differ, it is an ellipse.
Double angles and compound angles
If one equation contains 2t, expand it with a double angle formula first. Choose the form that matches the other equation: if x = cos t, use cos 2t = 2 cos²t − 1; if x = sin t, use cos 2t = 1 − 2 sin²t. Two trig equations can produce a curve with no trig in it at all, such as a parabola.
Which part of the curve is drawn?
The Cartesian equation describes the whole curve, but the parametric equations may only trace part of it. After eliminating t, ask what values x and y can actually take: feed the range of t through each equation.
Watch for expressions that can never be negative, such as t² or √t, and for endpoints of the interval for t. These restrict the domain and range, and examiners report that many students leave this step out.
Intersections and axis crossings
To find where a parametric curve meets a line or another curve, substitute the parametric expressions into the other equation and solve for t, then find the coordinates. For axis crossings, set x = 0 or y = 0, solve for t, then find the other coordinate. Factorise rather than divide by t, or you can lose a solution.
Worked examples
Example 1. A curve has parametric equations x = 2t, y = t². Find a Cartesian equation.
- x = 2t is the simpler equation, so t = x/2.
- Substitute: y = (x/2)² = x²/4.
- Check with t = 3: x = 6, y = 9, and 6²/4 = 9.
Answer: y = x²/4, a parabola.
Marks: M1 for rearranging for t, A1 for substituting and simplifying.
Example 2. A curve has parametric equations x = 3 cos t, y = 3 sin t, for 0 ≤ t < 2π. Find a Cartesian equation and describe the curve.
- Make the trig functions the subject: cos t = x/3 and sin t = y/3.
- cos and sin, so square and add: x²/9 + y²/9 = cos²t + sin²t = 1.
- Multiply by 9: x² + y² = 9.
Answer: x² + y² = 9, a circle with centre (0, 0) and radius 3.
Marks: M1 for making the trig functions the subject, M1 for squaring and adding, A1 for the equation and description.
Example 3. A curve has parametric equations x = cos t, y = cos 2t, for 0 ≤ t ≤ π. Find a Cartesian equation.
- The other equation is x = cos t, so use the all-cosine form: cos 2t = 2 cos²t − 1.
- So y = 2 cos²t − 1 = 2x² − 1.
- Check with t = π/3: x = ½ and y = cos(2π/3) = −½, and 2(½)² − 1 = −½.
Answer: y = 2x² − 1.
Marks: M1 for choosing the right form, A1 for substituting.
Example 4. The curve C has parametric equations x = 3t, y = t² − 1. Find where C meets the line y = x − 1.
- Substitute: t² − 1 = 3t − 1, so t² = 3t.
- Factorise, without dividing by t: t(t − 3) = 0, so t = 0 or t = 3.
- t = 0 gives (0, −1); t = 3 gives (9, 8).
- Both satisfy y = x − 1.
Answer: (0, −1) and (9, 8).
Marks: M1 for substituting, M1 for solving for t, A1 for both points.
Common mistakes
- Choosing the harder equation to rearrange. Rearrange the equation where t appears most simply, usually to the first power.
- Using the wrong identity. cos and sin need sin²t + cos²t = 1; sec and tan need sec²t = 1 + tan²t.
- Squaring before making the trig function the subject. From x = 3 cos t, write cos t = x/3 first, then square.
- Ignoring the domain and range. State which values of x and y the parametric curve actually reaches.
- Dividing by t. Factorise instead, so a solution such as t = 0 is not lost.
How parametric equations are examined
Parametric equations are Pure 4 content in Edexcel International A-Level (paper WMA14), in the coordinate geometry chapter, and second-year Pure in Edexcel A-Level (9MA0). Typical questions ask you to:
- show that a parametric curve has a given Cartesian equation
- state the domain and range of the curve, and explain which part is drawn
- find where a parametric curve meets a line or an axis
- write a circle or other curve in parametric form, or model motion with linear parametric equations
Exam-style questions with mark schemes are in the Pure 4 coordinate geometry exam questions. The Pure 4 coordinate geometry worksheet and the coordinate geometry revision engine give further practice.
Next, parametric differentiation finds gradients of these curves. The trig identities used here are covered in sec, cosec and cot and the double angle formulae.
Frequently asked questions
How do you convert parametric equations to Cartesian form?
Eliminate the parameter. Rearrange one equation for t and substitute into the other, or, if both are trig functions, use an identity such as sin²t + cos²t = 1.
Why does the domain matter for a parametric curve?
Because the parameter's range can restrict which values x and y take, so the parametric curve may be only part of the Cartesian curve.
What do x = r cos t and y = r sin t describe?
A circle with centre (0, 0) and radius r. Adding constants, x = a + r cos t and y = b + r sin t, moves the centre to (a, b).