91Ó°ÊÓ

Trigonometric identities are equations that relate the trigonometric functions such as sine, cosine, tangent, secant, etc., to one another. In this scenario, the identity \( \sec^2 t = 1 + \tan^2 t \) is particularly useful. This identity stems from the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \), which states that the sum of the squares of sine and cosine equals one. By dividing through by \( \cos^2 t \), we transform this into \( 1 + \tan^2 t = \sec^2 t \). This identity allows us to connect the functions in a way that eliminates the trigonometric parameter. Understanding these relationships helps in converting parametric equations into their Cartesian form, facilitating the sketching of graphs.

Eliminating a parameter means to find an equation for a curve that does not involve the parameter itself, giving a direct relationship between the variables. In our exercise, we have the parametric equations \( x = \sec t \) and \( y = \tan t \). By using the trigonometric identity \( \sec^2 t = 1 + \tan^2 t \), we can link \( x \) and \( y \) directly. Substitute \( x \) for \( \sec t \) into the identity to get \( x^2 = 1 + y^2 \). Simplifying this equation by solving for \( y^2 \), we find the Cartesian equation \( y^2 = x^2 - 1 \). This is now free of the parameter \( t \) and represents the same curve in the Cartesian coordinate system, allowing us to plot it without reference to \( t \).

A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations that guarantee a particular shape. The standard form of a hyperbola's equation is either \( x^2 - y^2 = 1 \) or \( y^2 - x^2 = 1 \), depending on its orientation. From the elimination of parameters, we derived the relationship \( y^2 = x^2 - 1 \). This equation describes a hyperbola that opens horizontally. The hyperbola has two branches, but the specific range of \( t \) indicates which part of the hyperbola to sketch. The central characteristic of hyperbolas is their asymptotic lines, which they get infinitely close to but never intersect. Understanding this property is crucial for sketching and analyzing hyperbolic graphs.

The direction of the parameter describes how the parametric equation traces the curve as the parameter changes. For \( t \) in the interval \( \pi \leq t < 3\pi/2 \), both \( x = \sec t \) and \( y = \tan t \) change as \( t \) varies. Within this range, \( \sec t \) passes from negative infinity towards zero, while \( \tan t \) moves from negative infinity to positive infinity. The effect is that the curve is traced in a specific direction from left to right. This is crucial for understanding motion or directionality on the graph, as it informs whether the curve is traced forwards or backwards as \( t \) increases. To sketch this properly, you should mark the direction with arrows indicating increasing \( t \). This aspect helps visualize dynamic processes represented by such curves, adding a layer of interpretation beyond merely plotting points.