Problem 20 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sec t, \quad y=\tan ^{2} t, \quad 0 \leq t<\pi / 2 $$

Step-by-Step Solution

Answer

The curve is a parabola given by \( y = x^2 - 1 \), defined for \( x \geq 1 \).

Step 1: Understand the Parametric Equations

We start with the given parametric equations: \( x = \sec t \) and \( y = \tan^2 t \). The parameter \( t \) is defined in the interval \( 0 \leq t < \pi/2 \). This is important for understanding the characteristics of the curve such as its domain and range.

Step 2: Sketch the Curve

To sketch the curve, evaluate the parametric equations at different values of \( t \) within the interval \([0, \pi/2)\). For example, at \( t = 0 \), \( x = \sec 0 = 1 \) and \( y = \tan^2 0 = 0 \). As \( t \) increases towards \( \pi/2 \), \( x \) approaches infinity as \( \sec t \) approaches infinity, and \( y \) also approaches infinity as \( \tan^2 t \) approaches infinity. This indicates a curve starting at (1,0) and extending to the right and upwards.

Step 3: Eliminate the Parameter

To find the rectangular equation, eliminate \( t \) from the parametric equations. We know \( x = \sec t \), which is \( x = \frac{1}{\cos t} \), so \( \cos t = \frac{1}{x} \). Also, since \( y = \tan^2 t \) and \( \tan t = \frac{\sin t}{\cos t} \), we have \( \tan^2 t = \frac{\sin^2 t}{\cos^2 t} \). Knowing that \( \sin^2 t = 1-\cos^2 t \), we can substitute to find \( \tan^2 t = \frac{1-\cos^2 t}{\cos^2 t} = \sec^2 t - 1 \). Therefore, \( y = x^2 - 1 \).

Step 4: Verify the Solution

Substitute the expression for \( y = x^2 - 1 \) back into the parametric equations to ensure consistency. For a given \( x \), find \( t \) using \( \cos t = \frac{1}{x} \) and compute \( y \) using \( \tan^2 t = \sec^2 t - 1 = x^2 - 1 \). This confirms that the rectangular equation \( y = x^2 - 1 \) correctly models the parametric equations within the defined interval.

Key Concepts

Rectangular-Coordinate Equation

When dealing with parametric equations, transforming them into a rectangular or Cartesian coordinate equation is a valuable step. It allows us to express the relationship between variables in terms of coordinates on a plane, which is often easier to interpret and visualize. In this exercise, we started with the parametric equations:

• \( x = \sec t \)
• \( y = \tan^2 t \)

Our goal was to convert these into a single equation relating \( x \) and \( y \). We eliminated the parameter \( t \) by leveraging trigonometric identities.
By recognizing that \( x = \frac{1}{\cos t} \), allowing us to express \( \cos t \) as \( \frac{1}{x} \). Then, expressing the \( \tan t \) in terms of sine and cosine and using the identity \( \sin^2 t = 1 - \cos^2 t \), we found :

• \( y = \tan^2 t = \sec^2 t - 1 = x^2 - 1 \)

Now, with the rectangular-coordinate equation, \( y = x^2 - 1 \), we effectively described the same curve using coordinates in a rectangular system.

Curve Sketching

Sketching the graph of a set of parametric equations can reveal the behavior of the curve and provide insights into its characteristics. For the given parametric equations, we have:

• \( x = \sec t \)
• \( y = \tan^2 t \)

In the interval \( 0 \leq t < \pi/2 \), both \( \sec t \) and \( \tan^2 t \) show asymptotic behavior as \( t \) approaches \( \pi/2 \).
This implies that the curve begins at the point (1,0) when \( t = 0 \). As \( t \) increases, both \( x \) and \( y \) increase towards infinity, indicating an upward and rightward trajectory. By evaluating specific points, such as when \( t = \frac{\pi}{4} \), you might calculate intermediate points to add more detail to the sketch.
This approach helps visualize tendencies like symmetry and inform us about asymptotes or intercepts, reinforcing our understanding of the curve's behavior.

Eliminating Parameters

Eliminating parameters is a technique used to simplify parametric equations into a more standard form. In other words, it's about expressing one variable directly in terms of another. This is advantageous when simplifying a problem or attempting to better understand relationships between variables.
To eliminate the parameter \( t \) in our parametric equations:

• Start from \( x = \sec t \), so \( \cos t = \frac{1}{x} \).
• Next, from \( y = \tan^2 t \), note that \( \tan^2 t = \frac{\sin^2 t}{\cos^2 t} \).
• Using \( \sin^2 t = 1 - \cos^2 t \) results in \( y = x^2 - 1 \).

By performing these steps, with calculus and algebraic manipulation, we express the parametric form \( x, y \) into a single, cohesive equation. This method is commonly seen in converting parametric equations to Cartesian form and facilitates tasks like curve sketching and analysis.