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Introduction

In mathematics, parametric equations are a powerful tool for describing curves and surfaces. However, sometimes we need to eliminate the parameter to find the Cartesian equation of a curve. In this article, we will explore how to eliminate the parameter from a pair of parametric equations and find its Cartesian equation. We will also specify the domain and range of the equation using interval notation.

What are Parametric Equations?

Parametric equations are a pair of equations that describe the coordinates of a point on a curve or surface in terms of a parameter, usually denoted as t. The general form of parametric equations is:

x(t) = f(t) y(t) = g(t)

where f(t) and g(t) are functions of the parameter t.

Eliminating the Parameter

To eliminate the parameter, we need to find a relationship between x and y that does not involve the parameter t. This can be done by using trigonometric identities, algebraic manipulations, or other mathematical techniques.

Example: Eliminating the Parameter

Let's consider the following pair of parametric equations:

x(u) = 2 \* cos(u) y(u) = 3 \* sin(u)

Our goal is to eliminate the parameter u and find the Cartesian equation of the curve.

Step 1: Use a Trigonometric Identity

We can use the trigonometric identity sin^2(u) + cos^2(u) = 1 to eliminate the parameter u. Let's start by squaring both equations:

x^2(u) = 4 \* cos^2(u) y^2(u) = 9 \* sin^2(u)

Now, we can add the two equations:

x^2(u) + y^2(u) = 4 \* cos^2(u) + 9 \* sin^2(u)

Step 2: Simplify the Equation

Using the trigonometric identity sin^2(u) + cos^2(u) = 1, we can simplify the equation:

x^2(u) + y^2(u) = 4 \* (1 - sin^2(u)) + 9 \* sin^2(u)

Expanding and simplifying, we get:

x^2(u) + y^2(u) = 4 - 4 \* sin^2(u) + 9 \* sin^2(u)

Combine like terms:

x^2(u) + y^2(u) = 4 + 5 \* sin^2(u)

Step 3: Eliminate the Parameter

Now, we can eliminate the parameter u by using the fact that sin^2(u) = (y^2(u)) / (3^2):

x^2(u) + y^2(u) = 4 + 5 \* ((y^2(u)) / (3^2))

Simplifying, we get:

x^2(u) + y^2(u) = 4 + (5/9) \* y^2(u)

Rearranging, we get:

x^2(u) = 4 - (4/9) \* y^2(u)

The Cartesian Equation

The Cartesian equation of the curve is:

x^2 = 4 - (4/9) \* y^2

Domain and Range

To find the domain and range of the equation, we need to consider the values of x and y that satisfy the equation.

Domain

The domain of the equation is the set of all possible values of x and y that satisfy the equation. Since x^2 = 4 - (4/9) \* y^2, we can see that x^2 is always non-negative. Therefore, the domain of the equation is:

x ∈ (-∞, ∞) y ∈ (-3, 3)

Range

The range of the equation is the set of all possible values of x and y that satisfy the equation. Since x^2 = 4 - (4/9) \* y^2, we can see that x^2 is always non-negative. Therefore, the range of the equation is:

x ∈ (-∞, ∞) y ∈ (-3, 3)

Conclusion

Q: What is the purpose of eliminating the parameter in parametric equations?

A: The purpose of eliminating the parameter is to find the Cartesian equation of a curve or surface, which is a more familiar and intuitive way of representing the curve or surface.

Q: How do I know when to use trigonometric identities to eliminate the parameter?

A: You can use trigonometric identities to eliminate the parameter when the parametric equations involve trigonometric functions, such as sine and cosine.

Q: What are some common trigonometric identities that can be used to eliminate the parameter?

A: Some common trigonometric identities that can be used to eliminate the parameter include:

  • sin^2(u) + cos^2(u) = 1
  • tan(u) = sin(u) / cos(u)
  • cot(u) = cos(u) / sin(u)

Q: How do I know when to use algebraic manipulations to eliminate the parameter?

A: You can use algebraic manipulations to eliminate the parameter when the parametric equations involve polynomial functions.

Q: What are some common algebraic manipulations that can be used to eliminate the parameter?

A: Some common algebraic manipulations that can be used to eliminate the parameter include:

  • Factoring
  • Expanding
  • Canceling common factors

Q: Can I use other mathematical techniques to eliminate the parameter?

A: Yes, you can use other mathematical techniques to eliminate the parameter, such as:

  • Using the chain rule to differentiate the parametric equations
  • Using the fundamental theorem of calculus to integrate the parametric equations

Q: How do I know when to use the chain rule to eliminate the parameter?

A: You can use the chain rule to eliminate the parameter when the parametric equations involve composite functions.

Q: What are some common applications of eliminating the parameter?

A: Some common applications of eliminating the parameter include:

  • Finding the Cartesian equation of a curve or surface
  • Determining the domain and range of a curve or surface
  • Solving optimization problems

Q: Can I use a calculator or computer software to eliminate the parameter?

A: Yes, you can use a calculator or computer software to eliminate the parameter, such as:

  • Graphing calculators
  • Computer algebra systems (CAS)
  • Programming languages (e.g. Python, MATLAB)

Q: How do I know when to use a calculator or computer software to eliminate the parameter?

A: You can use a calculator or computer software to eliminate the parameter when:

  • The parametric equations are complex or difficult to manipulate by hand
  • You need to find the Cartesian equation of a curve or surface quickly or accurately
  • You need to visualize the curve or surface

Conclusion

In this article, we have answered some common questions about eliminating the parameter from parametric equations. We have discussed the purpose of eliminating the parameter, common trigonometric identities and algebraic manipulations that can be used to eliminate the parameter, and common applications of eliminating the parameter. We have also discussed the use of calculators and computer software to eliminate the parameter.