Problem 21 - Precalculus: Mathematical for Calculus

Question

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. $$x=\cos ^{2} t, \quad y=\sin ^{2} t$$

Step-by-Step Solution

Answer

The curve is a line segment from (0, 1) to (1, 0), represented by x + y = 1.

Step 1: Understanding the Parametric Equations

The given parametric equations are \( x = \cos^2 t \) and \( y = \sin^2 t \). These equations express \( x \) and \( y \) in terms of the parameter \( t \). Our goal is to sketch the curve and find its rectangular-coordinate (Cartesian) equation.

Step 2: Sketching the Curve

To sketch the curve, recall the identities \( \cos^2 t + \sin^2 t = 1 \) for any angle \( t \). Since \( x = \cos^2 t \) and \( y = \sin^2 t \), substitute these into the identity: \( x + y = \cos^2 t + \sin^2 t = 1 \). Hence, the curve is the portion of the line \( x + y = 1 \) where both \( x \) and \( y \) are non-negative. This portion is a segment from \((0, 1)\) to \((1, 0)\).

Step 3: Eliminating the Parameter to Find Rectangular Equation

From Step 2, we know \( x + y = 1 \). This equation is derived by eliminating the parameter \( t \) using the identity mentioned. Therefore, the rectangular-coordinate equation of the curve is simply \( x + y = 1 \).

Step 4: Confirming the Curve and Its Domain

Since \( x = \cos^2 t \) and \( y = \sin^2 t \) are both squares of trigonometric functions, \( x, y \geq 0 \). Additionally, \( x + y = 1 \) confirms that the graph is a line segment from \((0, 1)\) to \((1, 0)\), representing the curve on the first quadrant.

Key Concepts

Curve Sketching

Curve sketching with parametric equations involves visualizing the path traced by a pair of equations as the parameter changes. In our example, the parametric equations are given by \( x = \cos^2 t \) and \( y = \sin^2 t \). When sketching curves with these equations, it is essential to understand the nature of the trigonometric functions involved.

To sketch the curve, first consider the trigonometric identity \( \cos^2 t + \sin^2 t = 1 \), which holds for any angle \( t \). This identity provides a clear path to sketching the curve.

• Set \( x = \cos^2 t \).
• Set \( y = \sin^2 t \).

Substitute these into the identity to find \( x + y = 1 \). Now, visualize this as a line within a coordinate plane. For our problem, the curve represents the portion of the line segment \( x + y = 1 \) in the first quadrant, running from the point \((0, 1)\) to \((1, 0)\). This part of the sketch is crucial as it shows the domain of the parameter as both \( x \) and \( y \) are non-negative, aligning perfectly with the given trigonometric equations.

Rectangular-Coordinate Equation

Switching from parametric to rectangular-coordinate equations involves eliminating the parameter and expressing the relationship directly between \( x \) and \( y \). For our given equations \( x = \cos^2 t \) and \( y = \sin^2 t \), we've already uncovered the key trigonometric identity \( \cos^2 t + \sin^2 t = 1 \).

By substituting \( x \) and \( y \) into this identity, we derive the rectangular-coordinate equation \( x + y = 1 \). This result provides a straightforward linear equation representing the curve when traced in a Cartesian plane.

• This equation is notably simple because it does not involve the parameter \( t \).
• It describes a linear path within the bounds established—here, a segment with endpoints at \((0, 1)\) and \((1, 0)\).

Thus, converting parametric equations into a single rectangular-coordinate equation simplifies recognizing the geometric shape of the curve, especially when parameters are related through well-known identities.

Trigonometric Identities

Trigonometric identities like \( \cos^2 t + \sin^2 t = 1 \) play a pivotal role in simplifying parametric equations and revealing the nature of the curve. These identities help us express one function in terms of another and are essential for transforming parametric equations into rectangular-coordinate equations.

Understanding these identities enables clear communication of how specific trigonometric functions relate:

• They allow elimination of parameters by binding together the parameter-dependent expressions for \( x \) and \( y \).
• Such identities often hint at geometric constraints the equations embody, like circles or line segments in Cartesian coordinates.

By effectively applying the identity \( \cos^2 t + \sin^2 t = 1 \), the complexity reduces as it constrains the \( x \) and \( y \) relationship to adhere to simple algebraic forms, exemplifying the power of these powerful mathematical shortcuts in curve sketching and transformation.