Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation for each curve and describe the curve. $$x=3 \tan t, y=2 \sec t ; \text { for } t \text { in }\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The rectangular equation is \(\frac{y^2}{4} - \frac{x^2}{9} = 1\) and it describes a hyperbola centered at the origin.

Step 1: Convert Parametric Equations to Trigonometric Identities

To eliminate the parameter, use the trigonometric identity based on the given parametric equations. Consider the identities involving tangent and secant: \ \[ \sec^2 t - \tan^2 t = 1 \]. In our case: \ \[ \left(\frac{x}{3}\right)^2 = \tan^2 t \ \left(\frac{y}{2}\right)^2 = \sec^2 t \] \ Subtract the tangent equation from the secant equation to apply the identity.

Step 2: Apply the Trigonometric Identity

Using the trigonometric identity \(\sec^2 t - \tan^2 t = 1\), replace \(\tan^2 t\) and \(\sec^2 t\) with the squared forms from the equations: \ \[ \left(\frac{y}{2}\right)^2 - \left(\frac{x}{3}\right)^2 = 1 \] \ This simplifies to an equation without the parameter \(t\).

Step 3: Simplify to Standard Form of Conic Section

Simplify the equation: \ \[ \frac{y^2}{4} - \frac{x^2}{9} = 1 \] \ This equation is in the form of \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), which represents a hyperbola centered at the origin.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles and sides of triangles. They are especially useful for transforming parametric equations, which involve a third variable or parameter, into more simplified forms. One key identity is

• \( \sec^2 t - \tan^2 t = 1 \)

This identity connects the secant and tangent trigonometric functions, which helps in converting equations expressed in terms of parameter \( t \) into a more conventional algebraic form.
To use this identity, it's essential to express \( \tan t \) and \( \sec t \) in terms of known variables. For instance, when given \( x = 3 \tan t \) and \( y = 2 \sec t \), you can rearrange them as follows:

• \( \left(\frac{x}{3}\right)^2 = \tan^2 t \)
• \( \left(\frac{y}{2}\right)^2 = \sec^2 t \)

By substituting these into the identity and simplifying, you can eliminate the parameter \( t \), transitioning the expression into a rectangular form. This technique efficiently bridges the gap between parametric and rectangular equations.

Hyperbola

A hyperbola is a distinct type of conic section characterized by its two separate curves, known as branches. Each branch mirrors the other across the line of symmetry. Recognizable from its equation format:

• \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

This form signals the presence of a hyperbola centered at the origin.
Unlike ellipses, hyperbolas have their focal points situated outside each branch, defining two sets of symmetrical paths. They are produced when a double cone is intersected with a plane in a certain way that does not pass through the vertices. Mathematically, hyperbolas can be understood through transformations and shifts from this standard form, enabling the analysis of their properties such as:

• The length between branches, known as the transverse axis, measured along \( y \) in this case.
• The orientation of the hyperbola - vertical for \( y^2 \) precedence in the equation.

Understanding hyperbolas aids in recognizing curves that naturally arise in response to equations like \( \frac{y^2}{4} - \frac{x^2}{9} = 1 \), marking it as a hyperbola.

Rectangular Equation

Rectangular equations are algebraic expressions that remove parametric dependencies, relying solely on \( x \) and \( y \) coordinates. For students, learning to convert parametric to rectangular form is crucial as it often simplifies analysis.
To achieve this, one must first identify relations, such as trigonometric identities, that can eliminate the parameter. In our exercise, the goal was to express \( x = 3 \tan t \) and \( y = 2 \sec t \) without relying on \( t \).

• Applying \( \sec^2 t - \tan^2 t = 1 \)
• Sub transforming it: \( \left(\frac{y}{2}\right)^2 - \left(\frac{x}{3}\right)^2 = 1 \)

These steps lead directly to a recognizable standard form of a hyperbola. Rectangular equations thus encapsulate the relationship between \( x \) and \( y \), enabling straightforward graphing and analysis. This method streamlines calculations, making it a vital aspect of algebra and trigonometry.