Problem 22 - Precalculus Mathematics for Calculus

Question

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 curve for \( 0 \leq t<\pi / 2 \) is \( y = x^2 - 1 \), with \( x \geq 1 \) and \( y \geq 0 \).

Step 1: Understand the Parametric Equations

We are given the parametric equations \( x = \sec t \) and \( y = \tan^2 t \). The parameter \( t \) varies from \( 0 \) to \( \frac{\pi}{2} \). Our task is to sketch this curve and eliminate the parameter \( t \) to find a rectangular equation.

Step 2: Sketch the Curve

To sketch the curve, note that as \( t \) approaches \( \frac{\pi}{2} \), \( \sec t \) (which is \( 1/\cos t \)) tends to infinity as \( \cos t \to 0 \). \( \tan t \) also tends towards infinity in this interval. The curve starts at (1,0) when \( t = 0 \) and moves toward the right, increasing infinitely as \( t \) approaches \( \frac{\pi}{2} \). The curve is in the positive \( x \)- and \( y \)-axis quadrant.

Step 3: Eliminate the Parameter

Use the trigonometric identity \( \sec^2 t = 1 + \tan^2 t \) to eliminate \( t \). Since \( x = \sec t \), we have \( x^2 = \sec^2 t \). Substitute \( x^2 \) for \( \sec^2 t \) in the identity: \( x^2 = 1 + \tan^2 t \). Substitute \( y = \tan^2 t \), so \( x^2 = 1 + y \). Thus, the rectangular equation is \( y = x^2 - 1 \).

Step 4: Determine the Domain and Range

Since \( t \) is restricted to \( 0 \leq t < \frac{\pi}{2} \), \( x \geq 1 \). As \( x = \sec t \) approaches infinity, the possible range for \( x \) is \([1, \infty)\). For \( y = x^2 - 1 \), the minimum value is \( 0 \) when \( x = 1 \). Therefore, the range for \( y \) is \([0, \infty)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables. They are vital when working with parametric equations as they allow for the transformation and simplification of expressions.
One of the most common ones is the Pythagorean identity, which in this case is used as:

• \( \sec^2 t = 1 + \tan^2 t \)

This identity is useful in our problem as it connects the two given parametric equations: \( x = \sec t \) and \( y = \tan^2 t \). By recognizing relationships like these, we can substitute parameters and solve for the rectangular coordinates. This simplification gives us more insight into the curve’s behavior in the Cartesian plane, transforming complex trigonometric expressions into easier-to-handle forms.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, are a way of representing points in the plane using pairs of numbers. In the context of parametric equations, converting from a parametric form (using \( t \)) to rectangular coordinates (using \( x \) and \( y \)) is a crucial step.
When an equation is in rectangular form, it is easier to interpret and graph. For example, when given the parametric equations \( x = \sec t \) and \( y = \tan^2 t \), we can initially identify how \( x \) and \( y \) depend on \( t \). By eliminating \( t \), we arrive at a more familiar equation, such as the one we derived: \( y = x^2 - 1 \). This relationship is easier to plot and analyze because it directly describes how \( y \) varies with \( x \). Understanding this transformation helps one leverage the strengths of each system: parametric for precision and detail, rectangular for clarity and simplicity.

Eliminating Parameters

Eliminating parameters in parametric equations means transforming them so the parameter \( t \) no longer explicitly appears, effectively converting them to rectangular equations. This process usually involves substitution using known identities or relationships between the variables.
In our problem, the elimination of the parameter \( t \) uses the trigonometric identity \( \sec^2 t = 1 + \tan^2 t \) to combine \( x = \sec t \) and \( y = \tan^2 t \) into the rectangular equation \( y = x^2 - 1 \). Here’s how it works:

• Substitute \( \sec^2 t \) with \( x^2 \), since \( x = \sec t \).
• Replace \( \tan^2 t \) with \( y \).

This transforms \( x^2 = 1 + \tan^2 t \) to \( x^2 = 1 + y \), hence deriving the rectangular form. This approach allows us to see and graph the entire curve more directly, understanding its full behavior without referencing \( t \). By understanding these methods, students can unlock the deeper geometric meanings behind parametric equations.