Problem 19 - 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. $$ x=\sec t, \quad y=\tan t, \quad 0 \leq t<\pi / 2 $$

Step-by-Step Solution

Answer

(a) The curve is a branch of a hyperbola in the first quadrant. (b) The rectangular equation is \(x^2 - y^2 = 1\).

Step 1: Understand the parametric equations

The given equations are \(x = \sec t\) and \(y = \tan t\). As \(t\) ranges from 0 to \(\pi/2\), both \(\sec t\) and \(\tan t\) are defined and increase from 1 to infinity.

Step 2: Sketch the curve using parametric equations

Plug in various values of \(t\) from 0 to just before \(\pi/2\), and calculate corresponding \(x\) and \(y\) values. Notice the trend: as \(t\) approaches \(\pi/2\), both \(x\) and \(y\) approach infinity. Plot these points on a coordinate plane to sketch the curve.

Step 3: Identify the trigonometric identity

Remember the trigonometric identity \(\sec^2 t = 1 + \tan^2 t\). This will help us eliminate the parameter \(t\) from the equations.

Step 4: Eliminate the parameter

Substitute \(x = \sec t\) and \(y = \tan t\) into the identity: \(x^2 = \sec^2 t = 1 + \tan^2 t = 1 + y^2\). Rearrange to get the rectangular equation: \(x^2 = y^2 + 1\).

Step 5: Rearrange the rectangular equation

The rectangular coordinate equation is \(x^2 - y^2 = 1\). This represents a hyperbola.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions. These equations hold true for all values of the variables involved. They are essential tools in eliminating parameters in parametric equations. In this exercise, we use the identity \( \sec^2 t = 1 + \tan^2 t \). This identity relates the secant and tangent functions which are fundamental in trigonometry.

Understanding trigonometric identities helps in transforming parametric equations into rectangular coordinates. This transformation is crucial because it allows us to represent curves in the simple and familiar Cartesian plane using \( x \) and \( y \) coordinates instead of a parameter such as \( t \). Thus, mastering trigonometric identities enables you to manipulate and solve complex trigonometric equations easily.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, are used to specify locations on the two-dimensional plane. They employ two values, \( x \) and \( y \), often plotted at a right angle to each other. These coordinates offer a straightforward way to work with curves.

Converting parametric equations to rectangular coordinates involves eliminating the parameter, in this case \( t \), using identities like \( \sec^2 t = 1 + \tan^2 t \). This allows us to create an equation like \( x^2 - y^2 = 1 \), which represents a specific curve in rectangular form. This conversion often simplifies further calculations and manipulations, such as analyzing or sketching the curve.

Hyperbola

A hyperbola is a type of smooth curve lying on a plane, defined by a specific kind of relationship between its coordinates. It is depicted by an equation in the form \( x^2 - y^2 = c \), where \( c \) is a constant. For the equation derived from our parametric equations, \( x^2 - y^2 = 1 \), it represents a hyperbola centered at the origin.

Hyperbolas are important in mathematics due to their unique properties, such as having two separate, mirror-image branches. In rectangular coordinates, a hyperbola appears as two distinct curves rather than a single, closed loop like an ellipse. Understanding their form and how to derive them from parametric equations is invaluable in advanced graph sketching and analysis.

Curve Sketching

Curve sketching is the process of drawing a graph that represents a particular set of equations. With parametric equations, sketching often involves plotting points for a range of parameter values. In this case, you test values of \( t \) between \( 0 \) and \( \pi/2 \) to find corresponding \( x \) and \( y \) coordinates. Point plotting gives you visual insights into how the curve behaves.

The fundamental goal of curve sketching is to understand the curve's shape and features. It includes identifying regions where the curve increases, decreases, approaches infinity, or intersects axes. In our exercise, as \( t \) approaches \( \pi/2 \), both \( x \) and \( y \) tend toward infinity, creating the asymptotic nature of the hyperbola's arms. Thorough sketching of curves provides a foundation for further mathematical analysis and understanding of their properties.