In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.


x = 5 sec t, y = 3 tan t

Verified step by step guidance

1. Start with the given parametric equations: \(x = 5 \sec t\) and \(y = 3 \tan t\).
2. Recall the fundamental trigonometric identity: \(\sec^2 t - \tan^2 t = 1\).
3. Express \(\sec t\) and \(\tan t\) in terms of \(x\) and \(y\) by isolating them from the parametric equations: \(\sec t = \frac{x}{5}\) and \(\tan t = \frac{y}{3}\).
4. Substitute these expressions into the identity to eliminate the parameter \(t\): \(\left(\frac{x}{5}\right)^2 - \left(\frac{y}{3}\right)^2 = 1\).
5. Simplify the equation to get the rectangular form of the curve: \(\frac{x^2}{25} - \frac{y^2}{9} = 1\). This represents a hyperbola. To sketch, plot this hyperbola and use the parametric definitions to determine the direction of increasing \(t\) by considering the signs of \(\sec t\) and \(\tan t\).

Key Concepts

Parametric Equations and Parameter Elimination

Parametric equations express coordinates (x, y) as functions of a parameter t. Eliminating the parameter involves rewriting these equations to form a single relationship between x and y, removing t. This process helps convert parametric forms into rectangular (Cartesian) equations for easier analysis and graphing.

Trigonometric Identities

Trigonometric identities, such as sec²t - tan²t = 1, are essential tools for eliminating parameters involving trigonometric functions. These identities allow substitution and simplification, enabling the conversion of parametric equations into a rectangular form by relating sec t and tan t without the parameter t.

Curve Sketching and Orientation

Sketching the curve involves plotting the rectangular equation and indicating the direction of increasing parameter t with arrows. Understanding orientation helps visualize how the curve is traced as t changes, which is important for interpreting motion or behavior of the curve in applications.