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8.2PureStretch

Using Trigonometric Identities

When parametric equations involve sin and cos, you cannot simply make t the subject. Instead, use a trigonometric identity — most often sin²t + cos²t = 1 — to eliminate the parameter.

26 min Parametric Equations
Edexcel A Level Maths: 8.2 Using Trigonometric IdentitiesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Convert trig parametric equations to Cartesian
  • Use sin²t + cos²t = 1
  • Recognise circle and ellipse forms
  • Apply other identities where needed
1

The Pythagorean route

If x=cos⁡tx = \cos t and y=sin⁡ty = \sin t, then x2+y2=cos⁡2t+sin⁡2t=1x^2 + y^2 = \cos^2 t + \sin^2 t = 1 — a unit circle. Rearrange the parametric equations to isolate cos⁡t\cos t and sin⁡t\sin t, then substitute into the identity.

sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1
The key identity for elimination.
Example 1Convert x=3cos⁡tx = 3\cos t, y=3sin⁡ty = 3\sin t to Cartesian form.
1cos⁡t=x3\cos t = \tfrac{x}{3}, sin⁡t=y3\sin t = \tfrac{y}{3}.
2(x3)2+(y3)2=1\left(\tfrac{x}{3}\right)^2 + \left(\tfrac{y}{3}\right)^2 = 1.
Answer
x2+y2=9x^2 + y^2 = 9 (a circle, radius 3).

Tip — Isolate cos t and sin t first, then square and add.

2

Other identities

Sometimes a double-angle or 1+tan⁡2t=sec⁡2t1 + \tan^2 t = \sec^2 t identity is the right tool. Choose the identity that matches the functions in the parametric equations.

Formula recap

sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1
Circle/ellipse elimination.
1+tan⁡2t=sec⁡2t1 + \tan^2 t = \sec^2 t
When tan/sec appear.

Common mistakes to avoid

Trying to make t the subject of a trig equation.
Use an identity to eliminate t instead.
Forgetting to square before adding.
sin²t + cos²t = 1 needs the squared terms.

Key takeaways

  • Use sin²t + cos²t = 1 to eliminate trig parameters.
  • Isolate cos t and sin t, then square and add.
  • x = r cos t, y = r sin t gives a circle of radius r.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. To eliminate t from x=cos⁡tx = \cos t, y=sin⁡ty = \sin t, use:

    • sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1
    • the sine rule
    • log laws
    • make t the subject
    Show answer and explanation

    Answer: sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1

    Pythagorean identity.

  2. x=3cos⁡tx = 3\cos t, y=3sin⁡ty = 3\sin t gives:

    • x2+y2=9x^2 + y^2 = 9
    • x2+y2=3x^2 + y^2 = 3
    • x+y=3x + y = 3
    • x2+y2=1x^2 + y^2 = 1
    Show answer and explanation

    Answer: x2+y2=9x^2 + y^2 = 9

    Circle radius 3.

  3. x=cos⁡tx = \cos t, y=sin⁡ty = \sin t traces a:

    • unit circle
    • parabola
    • line
    • hyperbola
    Show answer and explanation

    Answer: unit circle

    x²+y²=1.

  4. If tan t and sec t appear, the useful identity is:

    • 1+tan⁡2t=sec⁡2t1 + \tan^2 t = \sec^2 t
    • sin⁡2t+cos⁡2t=1\sin^2 t + \cos^2 t = 1
    • sin⁡2t\sin 2t
    • none
    Show answer and explanation

    Answer: 1+tan⁡2t=sec⁡2t1 + \tan^2 t = \sec^2 t

    Matches the functions.

  5. The first step is usually to:

    • isolate cos t and sin t
    • differentiate
    • integrate
    • plot points
    Show answer and explanation

    Answer: isolate cos t and sin t

    Then apply the identity.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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