91Ó°ÊÓ

A plane curve is a curve that lies flat on a plane. You can think of it as a path traced by a moving point. In the provided exercise, the curve is determined by the parametric equations \(x(t) = \text{csc}(t)\) and \(y(t) = \text{cot}(t)\) for \(t\) in the interval \(\frac{\text{Ï€}}{4} \to \frac{\text{Ï€}}{2}\). A plane curve generally provides a clear visual representation of a relationship between two variables. By graphing the instructed points, the relation between \(x\) and \(y\) becomes clear, showing the shape of the curve on the plane. This graphical method allows students to better understand how changes in one variable affect the other within the given range.

A rectangular equation, also known as a Cartesian equation, relates \(x\) and \(y\) directly in the form \(y = f(x)\). This form is typically easier to interpret and graph. To find the rectangular equation from the given parametric equations, we needed to eliminate the parameter \(t\). By recalling trigonometric identities and manipulating the given equations:
1. We identified \( \text{sin}(t) \) and \( \text{cos}(t) \) in terms of \(x\).
2. We used these identities to derive a direct relationship between \(x\) and \(y\): \( y = x \text{sqrt}(1 - \frac{1}{x^2})\). This rectangular equation helps in understanding the plane curve without dealing with the parameter \(t\).

Curve orientation indicates the direction in which the curve is traced as the parameter \(t\) increases. For the given parametric equations, as \(t\) moves from \(\frac{\text{π}}{4}\) to \(\frac{\text{π}}{2}\), the curve points shift from \((\text{√}2, 1)\) to \((1, 0)\). Understanding the curve's orientation is crucial because it tells us about the progression along the curve. It also aids in estimating the behavior of the curve between plotted points. By marking the direction with arrows on the graph, we visually depict how the parameter influences movement along the curve.

Converting parametric equations to a rectangular form involves eliminating the parameter (in this case \(t\)) to relate \(x\) directly to \(y\). Here's how we did it for the exercise:
1. We noted \( x(t) = \text{csc}(t) = \frac{1}{\text{sin}(t)} \), and therefore, \( \text{sin}(t) = \frac{1}{x} \).
2. We then substituted this into the second equation \( y(t) = \text{cot}(t) = \frac{\text{cos}(t)}{\text{sin}(t)} \).
3. Using \( \text{cos}(t) = \text{sqrt}(1 - \text{sin}^2(t)) \), we found \( \text{cos}(t) = \text{sqrt}(1 - \frac{1}{x^2}) \).
4. Hence, the equation became \( y = x \text{sqrt}(1 - \frac{1}{x^2}) \). This process often involves trigonometric identities and algebraic manipulation, but it's quintessential for understanding how the plane curve behaves without the parametric dependency.