Eliminating the Parameter

Introduction to Eliminating the Parameter

In trigonometry and precalculus, parametric equations express the coordinates of points on a curve as functions of a parameter, typically denoted as t. To analyze or graph these curves more easily, it is often useful to eliminate the parameter and rewrite the equations in terms of only x and y, resulting in a rectangular (Cartesian) equation.

• Parametric Equations: Equations where both x and y are given in terms of a third variable (the parameter), usually t.

• Rectangular Equation: An equation involving only x and y, with the parameter eliminated.

General Method for Eliminating the Parameter

Given parametric equations:

To eliminate the parameter:

1. Solve one equation for t in terms of x or y.

2. Substitute this expression into the other equation to obtain a relationship between x and y.

Example:

Solve for t from the first equation:

Substitute into the second equation:

Graphing Parametric Equations and Their Rectangular Forms

To visualize the relationship, plot points for various values of t and connect them to form the curve. After eliminating the parameter, the resulting rectangular equation can be graphed using standard Cartesian methods.

• When the parameter t is restricted to a certain interval, the graph may represent only a portion of the full rectangular curve.

Practice Example

Solve for t from the first equation:

Substitute into the second equation:

Eliminating the Parameter with Trigonometric Functions

Equations Involving Sine and Cosine

When parametric equations involve trigonometric functions, such as sine and cosine, eliminating the parameter often requires using trigonometric identities.

• Given: ,

• Use the Pythagorean identity:

Example:

Express and in terms of and :

Substitute into the identity:

This is the equation of an ellipse.

Practice Example with Trigonometric Functions

Eliminate the parameter:

Substitute into the identity:

Summary Table: Common Parametric to Rectangular Conversions

Key Points

• Eliminating the parameter simplifies analysis and graphing of curves.

• For trigonometric parametric equations, use identities to relate x and y.

• Always consider the domain of the parameter, as it may restrict the portion of the curve represented.

Additional info: These techniques are foundational for understanding conic sections, motion in the plane, and applications in physics and engineering.