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 Video by Zeeshan Zamurred Parametric Equations

Edexcel A Level Maths: 8.2 Using Trigonometric IdentitiesWatch the full walkthrough 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

The Pythagorean route

If $x = cos t$ and $y = sin t$ , then $x^{2} + y^{2} = cos^{2} t + sin^{2} t = 1$ — a unit circle. Rearrange the parametric equations to isolate $cos t$ and $sin t$ , then substitute into the identity.

sin^{2} t + cos^{2} t = 1

The key identity for elimination.

cos t = \frac{x}{3}

sin t = \frac{y}{3}

(\frac{x}{3})^{2} + (\frac{y}{3})^{2} = 1

Answer

x^{2} + y^{2} = 9

(a circle, radius 3).

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

Other identities

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

Formula recap

sin^{2} t + cos^{2} t = 1

Circle/ellipse elimination.

1 + 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.

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