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Worked solution

Eliminate trigonometric parameters to identify an ellipse

Find the Cartesian equation of x=8cos⁡θx=8\cos\theta, y=4sin⁡θy=4\sin\theta, 0≤θ≤2π0\le\theta\le2\pi.

CJC · Lecture · Example 5

Apply the sine-cosine identity

cos⁡θ=x/8\cos\theta=x/8 and sin⁡θ=y/4\sin\theta=y/4. Thus (x/8)2+(y/4)2=1(x/8)^2+(y/4)^2=1, or x2/64+y2/16=1x^2/64+y^2/16=1. The entire ellipse is traced counterclockwise, starting and ending at (8,0)(8,0), centred at (0,0)(0,0), with the other axial endpoints (−8,0)(-8,0) and (0,±4)(0,\pm4).

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Ellipse centred at the origin, horizontal endpoints (−8,0),(8,0), vertical endpoints (0,−4),(0,4), with increasing-theta counterclockwise direction.
Ellipse centred at the origin, horizontal endpoints (−8,0),(8,0), vertical endpoints (0,−4),(0,4), with increasing-theta counterclockwise direction.
Ellipse centred at the origin, horizontal endpoints (−8,0),(8,0), vertical endpoints (0,−4),(0,4), with increasing-theta counterclockwise direction.

Common mistakes

The denominators are the squared semi-axes, not 8 and 4 themselves.

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