Eliminating the Parameter in Parametric and Trigonometric Equations

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:

• $x = f(t)$

• $y = g(t)$

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:

• $x = t - 2$

• $y = 3t^2 - 2$

Solve for t from the first equation: $t = x + 2$

Substitute into the second equation:

$y = 3(x + 2)^2 - 2$

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

• $x(t) = 2t - 1$

• $y(t) = t^2 - 2$

Solve for t from the first equation: $t = \frac{x + 1}{2}$

Substitute into the second equation:

$y = \left(\frac{x + 1}{2}\right)^2 - 2$

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: $x = a \cos t$, $y = b \sin t$

• Use the Pythagorean identity: $\cos^2 t + \sin^2 t = 1$

Example:

• $x = 2 \cos t$

• $y = 3 \sin t$

Express $\cos t$ and $\sin t$ in terms of $x$ and $y$:

• $\cos t = \frac{x}{2}$

• $\sin t = \frac{y}{3}$

Substitute into the identity:

$\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1$

This is the equation of an ellipse.

Practice Example with Trigonometric Functions

• $x(t) = 3 \cos t$

• $y(t) = 2 \sin t$

• $0 \leq t \leq 2\pi$

Eliminate the parameter:

• $\cos t = \frac{x}{3}$

• $\sin t = \frac{y}{2}$

Substitute into the identity:

$\left(\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1$

Summary Table: Common Parametric to Rectangular Conversions

Parametric Form	Rectangular Equation	Curve Type
$x = t$, $y = t^2$	$y = x^2$	Parabola
$x = a \cos t$, $y = b \sin t$	$\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1$	Ellipse
$x = r \cos t$, $y = r \sin t$	$x^2 + y^2 = r^2$	Circle

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.