Problem 16 - Precalculus with Calculus Previews

Question

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=\tan t, y=\sec t,-\pi / 2

Step-by-Step Solution

Answer

The rectangular equation is \\(y^2 = 1 + x^2\\).

Step 1: Introduce Trigonometric Identity

Use the trigonometric identity \(1 + \tan^2 t = \sec^2 t\) to relate \(x\) and \(y\) without the parameter \(t\). This identity will help us eliminate the parameter.

Step 2: Substitute for Trigonometric Functions

Knowing that \(x = \tan t\) and \(y = \sec t\), it follows that \(1 + x^2 = y^2\) from the trigonometric identity. Here, we directly substitute \(\tan t\) and \(\sec t\) with \(x\) and \(y\) respectively.

Step 3: Formulate Rectangular Equation

The relationship \(1 + x^2 = y^2\) is a rectangular equation that represents the curve without the parameter \(t\). This equation is now in terms of \(x\) and \(y\) only.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools in mathematics, especially in trigonometry and calculus, since they allow us to simplify expressions and solve equations. In this exercise, we used a specific identity:

• \(1 + \tan^2 t = \sec^2 t\)

This identity is particularly useful because it connects two trigonometric functions, \(\tan t\) and \(\sec t\), into a single equation. By using this identity, we can relate \(x\) and \(y\) without involving the parameter \(t\). This step is crucial for eliminating the parameter, which simplifies the problem and makes it easier to analyze the relationship between \(x\) and \(y\).
It's also important to understand that these identities are not just random formulas; they are derived from the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\). By dividing each term by \(\cos^2 t\), you get the identity we used above. Recognizing and applying these identities correctly helps in transforming and solving trigonometric problems efficiently.

Rectangular Equation

A rectangular equation, also known as a Cartesian equation, expresses the relationship between two variables in a coordinate plane without using a parameter. It uses the standard \(x\) and \(y\) coordinates, which are familiar from graphing in algebra.
In this exercise, we started with the parametric equations \(x = \tan t\) and \(y = \sec t\). We then used the trigonometric identity to eliminate the parameter \(t\), allowing us to combine these into a rectangular equation:

• \(1 + x^2 = y^2\)

This equation is in the form of \(x\) and \(y\) only, making it a rectangular equation that describes the same graph as the original parametric equations.
Understanding rectangular equations is important because they provide a straightforward way to visualize functions and plot graphs, aiding in the comprehension of the relationships between variables. They are the backbone of geometry in the coordinate plane, portraying how changes in one variable affect another.

Eliminating Parameters

Eliminating parameters is the process of removing a parameter from parametric equations to obtain a single equation that relates the other variables directly. This is done to simplify the expression and transition from a parametric form to a more intuitive one, like the rectangular form.
In our example, the parametric equations involved \(x = \tan t\) and \(y = \sec t\). By using the identity \(1 + \tan^2 t = \sec^2 t\), we were able to eliminate the parameter \(t\). The identity allowed us to express \(y^2\) in terms of \(x^2\) as

• \(1 + x^2 = y^2\)

This move joins the equations into a single relationship between \(x\) and \(y\).

Parameter elimination is a key technique in calculus and algebra since it simplifies the analysis of curves and functions by removing the intermediary parameter. This approach offers a direct pathway to visualize data, making it possible to plot and interpret the function on a standard \(xy\)-graph. Plus, it often simplifies equations, making it easier to analyze and solve them.