Parametric Equations Math Example 2

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

Example 2

medium
Eliminate the parameter from x=3cos⁡(t)x = 3\cos(t) and y=3sin⁡(t)y = 3\sin(t).

Solution

  1. 1
    From the parametric equations: cos⁡(t)=x3\cos(t) = \frac{x}{3} and sin⁡(t)=y3\sin(t) = \frac{y}{3}.
  2. 2
    Use the Pythagorean identity cos⁡2(t)+sin⁡2(t)=1\cos^2(t) + \sin^2(t) = 1.
  3. 3
    Substitute: (x3)2+(y3)2=1\left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1, which gives x29+y29=1\frac{x^2}{9} + \frac{y^2}{9} = 1.
  4. 4
    Simplify: x2+y2=9x^2 + y^2 = 9.

Answer

x2+y2=9x^2 + y^2 = 9
When parametric equations involve sine and cosine, the Pythagorean identity is the key to eliminating the parameter. The result is a circle of radius 3 centered at the origin. The parameter tt represents the angle, tracing the circle counterclockwise as tt increases.

About Parametric Equations

A way of defining a curve by expressing both xx and yy as separate functions of a third variable (parameter), typically tt: x=f(t)x = f(t), y=g(t)y = g(t).

Learn more about Parametric Equations →

More Parametric Equations Examples

Example 1 easy

Eliminate the parameter from [formula] and [formula] to find the rectangular equation.

Example 3 medium

Find parametric equations for the line through [formula] and [formula].

Example 4 hard

Eliminate the parameter from [formula] and [formula] and describe the curve.

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