Problem 17 - 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. $$x=\sec t, \quad y=\tan t, \quad 0 \leq t<\pi / 2$$

Step-by-Step Solution

Answer

The curve is the right branch of a hyperbola, represented by \( x^2 - y^2 = 1 \).

Step 1: Understanding the Parametric Equations

The given parametric equations are \( x = \sec t \) and \( y = \tan t \) with the parameter \( t \) varying from \( 0 \) to \( \pi/2 \). The task is to understand these equations in terms of the trigonometric identities: \( \sec t = \frac{1}{\cos t} \) and \( \tan t = \frac{\sin t}{\cos t} \).

Step 2: Sketching the Curve

To sketch the curve, evaluate the points for different values of \( t \) within the interval \( 0 \leq t < \pi/2 \). At \( t = 0 \), \( x = 1 \) and \( y = 0 \). As \( t \) approaches \( \pi/2 \), \( x \) approaches infinity and \( y \) also approaches infinity. The curve forms the right branch of a hyperbola extending to the first quadrant.

Step 3: Eliminate the Parameter to Find Rectangular Equation

Use the identity \( 1 + \tan^2 t = \sec^2 t \) to eliminate \( t \). Substituting the given parametric equations gives \( 1 + y^2 = x^2 \). Thus, the rectangular-coordinate equation becomes \( x^2 - y^2 = 1 \).

Step 4: Finalizing the Rectangular Equation

The derived rectangular equation \( x^2 - y^2 = 1 \) represents a hyperbola, consistent with the curve sketch. This equation is the rectangular representation of the given parametric equations.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations representing relationships among trigonometric functions. They're pivotal when working with parametric equations, as they help transform and simplify expressions to reveal underlying relationships. In our problem, we use two key trigonometric identities:

• The secant identity: \( \sec t = \frac{1}{\cos t} \)
• The tangent identity: \( \tan t = \frac{\sin t}{\cos t} \)

These identities help convert trigonometric expressions into equivalent forms that can be more easily manipulated. Particularly helpful is using the Pythagorean identity: \( 1 + \tan^2 t = \sec^2 t \). This identity allows us to eliminate the parameter \( t \) from parametric equations. When you substitute these identities into the problem, you streamline transforming the parametric form of equations into a simpler rectangular form. It is like translating a roadmap (parametric form) into directions (rectangular form) for easier understanding.

Hyperbola

A hyperbola is a type of conic section that appears as a set of all points where the absolute difference of the distances to two fixed points, known as foci, is constant. The hyperbola in this exercise is derived from our simplified rectangular equation, \( x^2 - y^2 = 1 \), a standard form of a hyperbola centered at the origin. Here:

• The main axes of the hyperbola lie on the coordinate axes.
• The equation indicates a pair of diverging curves consisting of two branches, each resembling an open curve.

In the context of the parametric equations, the right branch is described by the given range of \( t \) and extends into the first quadrant. As \( t \) approaches \( \frac{\pi}{2} \), both \( x \) and \( y \) grow large, illustrating the characteristic stretching of a hyperbola's arms into infinity.

Rectangular-Coordinate Equation

A rectangular-coordinate equation is an expression that defines a curve using Cartesian coordinates \( x \) and \( y \). In this exercise, the task is to convert given parametric equations into a rectangular form. This involves removing the parameter \( t \) altogether. We achieved this by leveraging known identities, where we utilized \( 1 + \tan^2 t = \sec^2 t \) to eliminate \( t \), a process called parameter elimination. The result is the expression \( x^2 - y^2 = 1 \), showcasing a hyperbola in rectangular coordinates. This integration and transformation highlight how one can translate between different coordinate systems, providing a broader understanding of geometric shapes defined by equations. The transformation from parametric to rectangular not only simplifies working with the curve but also helps graphically visualize its shape on the standard coordinate plane without dealing with trigonometric parameters.