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Pure · Coordinate geometry

Chapter 1 · 4

The idea

Parametric curves and trig identities

Converting trigonometric parametric equations to Cartesian form using identities — sin²+cos²≡1 for circles and ellipses, sec²−tan²≡1, and the double-angle formulae — and recognising the curve that results.

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Pure · Coordinate geometry

Parametric curves and trig identities

Converting trigonometric parametric equations to Cartesian form using identities — sin²+cos²≡1 for circles and ellipses, sec²−tan²≡1, and the double-angle formulae — and recognising the curve that results.

Why it works

When you can't isolate t

When the parametric equations involve sin⁡t\sin t, cos⁡t\cos t (or sec⁡t\sec t, tan⁡t\tan t), you usually cannot isolate tt tidily. Instead you eliminate it with a trigonometric identity: arrange each equation so that a trig function is the subject, then feed those into an identity that the functions must satisfy. The identity does the eliminating that algebra can't.

Circles and ellipses from sin² + cos² ≡ 1

This is the workhorse. For x=acos⁡tx = a\cos t, y=bsin⁡ty = b\sin t:

cos⁡t=xa, sin⁡t=yb →cos⁡2+sin⁡2=1 x2a2+y2b2=1\cos t = \frac{x}{a},\ \sin t = \frac{y}{b} \ \xrightarrow{\cos^2 + \sin^2 = 1}\ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

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