Problem 20 - Precalculus: Mathematical 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 is \( y = x^2 - 1 \) for \( x \geq 1 \).

Step 1: Understanding the Parametric Equations

The parametric equations given are \( x = \sec t \) and \( y = \tan^2 t \). The parameter \( t \) represents an angle ranging from 0 to just under \( \frac{\pi}{2} \). \( \sec t \) is defined as \( \frac{1}{\cos t} \), and \( \tan t = \frac{\sin t}{\cos t} \), giving us \( \tan^2 t = \left( \frac{\sin t}{\cos t} \right)^2 \). We will use these expressions to sketch the curve and find a corresponding rectangular-coordinate equation.

Step 2: Sketching the Curve

To sketch the curve, observe the behavior of \( x = \sec t \) and \( y = \tan^2 t \) as \( t \) varies from 0 to just before \( \frac{\pi}{2} \). As \( t \) increases towards \( \frac{\pi}{2} \), \( \cos t \) approaches 0, making \( \sec t \) approach infinity. Simultaneously, \( \tan t \) also approaches infinity as \( \cos t \) approaches zero. Hence, both \( x \) and \( y \) increase without bound. The curve starts at \( (x, y) = (1, 0) \) when \( t = 0 \), tracing only in the first quadrant towards positive infinity.

Step 3: Eliminate the Parameter

To find a rectangular-coordinate equation, eliminate the parameter \( t \). Given \( x = \sec t = \frac{1}{\cos t} \), we have \( \cos t = \frac{1}{x} \). From the identity \( \tan^2 t = \sec^2 t - 1 \), we substitute \( \sec t = x \) to get \( y = \tan^2 t = x^2 - 1 \). Thus, the rectangular-coordinate equation representing the curve is \( y = x^2 - 1 \).

Step 4: Verify Domain of Rectangular Equation

Finally, verify the domain for the rectangular equation \( y = x^2 - 1 \). From the range \( 0 \leq t < \frac{\pi}{2} \), \( \sec t \geq 1 \), indicating \( x \geq 1 \). Therefore, the curve of the function \( y = x^2 - 1 \) is valid for \( x \geq 1 \) within the context of this problem.

Key Concepts

Rectangular-Coordinate Equation

To understand the rectangular-coordinate equation, we start by analyzing parametric equations that define a curve using a parameter, in this case, \( t \). Finding a rectangular-coordinate equation means transforming those parametric equations into one equation involving only \( x \) and \( y \). In our exercise, we were given \( x = \sec t \) and \( y = \tan^2 t \).
The goal is to eliminate the parameter \( t \). We know from trigonometric identities that \( \sec t = \frac{1}{\cos t} \) and \( \tan^2 t = \frac{\sin^2 t}{\cos^2 t} \), which relate these parametric forms to known trigonometric functions. By using the identity \( \tan^2 t = \sec^2 t - 1 \), substituting \( \sec t = x \) yields the rectangular-coordinate equation \( y = x^2 - 1 \).
This form provides a way to consider the relationship between \( x \) and \( y \) without needing \( t \), allowing for easier sketching and analysis of the curve on a standard coordinate system.

Curve Sketching

Curve sketching involves plotting the graph of an equation to understand its shape and direction. Using parametric equations like \( x = \sec t \) and \( y = \tan^2 t \), sketching begins by observing how the values of \( x \) and \( y \) change as \( t \) varies.
When \( t \) is 0, we start at the point \( (1, 0) \) since \( x = \sec 0 = 1 \) and \( y = \tan^2 0 = 0 \). As \( t \) approaches \( \frac{\pi}{2} \), both \( \sec t \) and \( \tan^2 t \) increase towards infinity because \( \cos t \) approaches zero. This causes both \( x \) and \( y \) values to grow larger without bound.
Hence, the curve remains in the first quadrant, starting at \( (1,0) \) and moving infinitely upwards and to the right as \( x \) increases. This sketch provides a visual representation making it easier to understand how these values interact, helping us grasp the underlying behavior of the curve.

Trigonometric Identities

Trigonometric identities are essential in mathematics, providing relationships between trigonometric functions that simplify the manipulation of these functions. In the given parametric equations \( x = \sec t \) and \( y = \tan^2 t \), we employed trigonometric identities to eliminate the parameter \( t \).
One key identity used is \( \tan^2 t = \sec^2 t - 1 \), which connects tangent to secant. Substituting \( \sec t = x \) directly into \( \tan^2 t \) helps derive the rectangular-coordinate equation \( y = x^2 - 1 \).
These identities are the backbone of converting parametric forms to rectangular equations, allowing relationships between variables to be expressed in simpler, more familiar forms. Understanding these identities enables the transformation of complex trigonometric relationships into widely used coordinate equations, aiding in sketching and analysis.