(a) Eliminate the parameter to find a Cartesian equation of...

(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
$x=\tan ^2 \theta, \quad y=\sec \theta, \quad-\pi / 2<\theta<\pi / 2$

Key Concepts

Graphing of Parametric Curves

Graphing parametric curves involves plotting the set of points defined by the parametric equations in the coordinate plane. Unlike standard Cartesian graphs, this method often better illustrates the path or trajectory of a point as the parameter varies, allowing for a clear visualization of the curve's overall behavior.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. In the context of eliminating parameters from equations that include functions like tangent and secant, identities such as sec^2? = 1 + tan^2? are crucial for rewriting the equations in Cartesian form.

Direction of Motion

The direction of motion on a parametric curve is indicated by the order in which points are plotted as the parameter increases. Arrows on the graph show the path that the curve follows, which is especially important in applications such as physics where the sense of motion provides additional context to the curve.

Parametric Equations

Parametric equations define a curve by expressing the coordinates of its points as functions of a variable (the parameter). This approach is useful for describing complex curves and motions that are difficult or impossible to represent with a single Cartesian equation.

Elimination of Parameter

Eliminating the parameter involves rewriting the system of parametric equations into a single equation relating x and y by removing the parameter. This is often achieved by solving one of the parametric equations for the parameter and substituting it into the other equation, or by using a known relation between the functions of the parameter.

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