Parametric Graphs Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Describe the graph of x=2cos⁡(t)x = 2\cos(t), y=5sin⁡(t)y = 5\sin(t) for 0≤t≤2π0 \le t \le 2\pi, including shape, direction, and starting point.

Solution

  1. 1
    Eliminate the parameter: x24+y225=cos⁡2t+sin⁡2t=1\frac{x^2}{4} + \frac{y^2}{25} = \cos^2 t + \sin^2 t = 1. This is an ellipse.
  2. 2
    At t=0t = 0: (x,y)=(2,0)(x, y) = (2, 0). At t=π2t = \frac{\pi}{2}: (0,5)(0, 5). At t=πt = \pi: (−2,0)(-2, 0). At t=3π2t = \frac{3\pi}{2}: (0,−5)(0, -5).
  3. 3
    The point moves counterclockwise around the ellipse, starting and ending at (2,0)(2, 0).
  4. 4
    The ellipse has semi-major axis 55 (vertical) and semi-minor axis 22 (horizontal).

Answer

Ellipse x24+y225=1, traced counterclockwise from (2,0)\text{Ellipse } \frac{x^2}{4} + \frac{y^2}{25} = 1 \text{, traced counterclockwise from } (2, 0)
Parametric equations with sine and cosine naturally describe ellipses (or circles when the coefficients are equal). The standard parameterization x=acos⁡tx = a\cos t, y=bsin⁡ty = b\sin t traces the ellipse counterclockwise. The parameter tt represents the eccentric anomaly, not the actual angle from the center.

About Parametric Graphs

Plotting and analyzing curves defined by parametric equations x=f(t)x = f(t), y=g(t)y = g(t), including eliminating the parameter, determining direction of motion, and finding tangent lines.

Learn more about Parametric Graphs →

More Parametric Graphs Examples

Example 1 easy

Sketch the direction of motion for the parametric curve [formula], [formula] as [formula] increases

Example 3 medium

What is the difference between the graphs of (a) [formula], [formula] and (b) [formula], [formula]?

Example 4 hard

Find the slope of the tangent line to the curve [formula], [formula] at the point where [formula].

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