Problem 22 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=\sec t, \quad y=\tan ^{2} t, \quad 0 \leq t<\pi / 2 $$

Step-by-Step Solution

Answer

The sketch shows a curve starting at (1, 0) rising upwards; the rectangular equation is \(y = x^2 - 1\).

Step 1: Understand the Parametric Equations

The given parametric equations are:\[ x = \sec t \] and \[ y = \tan^2 t \]. These describe a curve made up of points \((x, y)\) on a graph. We need to sketch this curve using the range provided \(0 \leq t < \pi/2\).

Step 2: Sketching the Curve

As \(t\) varies from \(0\) to \(\pi/2\), \(x = \sec t\) goes from 1 towards infinity since \(\sec t = 1/\cos t\) and \(\cos t\) approaches zero. \(y = \tan^2 t\) starts at 0 and increases towards infinity. This means the curve sketches upwards, starting from point (1, 0) and moving to infinity, following a path that reflects the behavior of a quadratic function starting at \((1,0)\) and rising steeply.

Step 3: Derive the Rectangular Coordinate Equation

To find a rectangular equation, eliminate the parameter \(t\). Using the identity \(\tan^2 t = \sec^2 t - 1\), we substitute \(x\) for \(\sec t\) in \(y = \tan^2 t\). Thus, \(y = x^2 - 1\). This is the rectangular equation matching the parametric description.

Key Concepts

Rectangular Coordinate Equation

Understanding a rectangular coordinate equation is essential when dealing with parametric equations. In parametric form, both x and y are expressed in terms of a third parameter, often t. To grasp the behavior of the curve defined by these parametric equations, we strive to find a rectangular equation where y is directly a function of x.

In this exercise, the parametric equations provided are:

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

Notice how each variable is described using the parameter \( t \). To transform these into a rectangular coordinate equation, we use known trigonometric identities. For example, from trigonometric identities, we know that \( \tan^2 t = \sec^2 t - 1 \). By replacing \( \sec t \) with \( x \) (as given in the parametric form), we can express \( y \) as \( y = x^2 - 1 \).

This equation \( y = x^2 - 1 \) is the rectangular representation of the parametric curve. Here, the x and y values now directly relate without the need for t, allowing us to graph the curve using standard Cartesian coordinates.

Eliminating the Parameter

Eliminating the parameter involves removing the parameter from the equations to obtain a direct relationship between x and y. This process aids in simplifying parametric curves into easily manageable rectangular forms.

In our given scenario:

• The parameter \( t \) is initially present in both \( x = \sec t \) and \( y = \tan^2 t \).
• To eliminate \( t \), we utilize the trigonometric identity \( \tan^2 t = \sec^2 t - 1 \).

By noting that \( x = \sec t \), it follows that \( \sec^2 t = x^2 \). Substituting this into the identity for \( \tan^2 t \) allows us to redefine \( y \) in terms of \( x \) alone, resulting in \( y = x^2 - 1 \).

This approach establishes a direct connection between the x and y coordinates, thereby eliminating the need for the parameter \( t \). The curve's characteristics can now be studied using the simple relation \( y = x^2 - 1 \). This method of parameter elimination is handy when converting parametric equations into their rectangular counterparts.

Graphing Parametric Curves

Graphing parametric curves involves plotting coordinates determined by a set of parametric equations over a defined range of the parameter.

For the parametric equations provided:

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

It’s necessary to track how x and y change as \( t \) moves through its allowed range, which is from 0 to \( \pi/2 \).

As \( t \) approaches \( \pi/2 \), \( \sec t \) (which is \( 1/\cos t \)) increases from 1 to infinity since \( \cos t \) approaches zero. Simultaneously, \( \tan^2 t \) starts at 0 and advances towards infinity. This correlation indicates the curve starts at point (1, 0) and ascends sharply into the top-right quadrant of the Cartesian plane.

By extracting the rectangular coordinate equation \( y = x^2 - 1 \), the characteristics of the curve like its concavity and direction can be easily understood and sketched. This process saves time and enhances our understanding of the curve's intricate dynamics.