Problem 17 - Calculus Early Transcendentals

Question

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=\cos t, y=\sin ^{2} t ; 0 \leq t \leq \pi$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter \(t\) from the parametric equations \(x=\cos t\) and \(y=\sin^2 t\), describe the curve represented, and indicate its positive orientation. Answer: The equation in \(x\) and \(y\) is \(x^2 + y = 1\). The curve represented is the upper half of a unit circle, and its positive orientation is counterclockwise.

Step 1: Eliminate the parameter \(t\)

Using the parametric equations given, \(x=\cos t\) and \(y=\sin^2 t\), we can eliminate the parameter \(t\) by using the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). Since we know \(x = \cos t\), we can solve for \(\cos^2 t\) by squaring \(x\), which gives us \(x^2 = \cos^2 t\). Similarly, we have \(y=\sin^2 t\). Now, substitute both equations into the trigonometric identity: $$x^2 + y = 1$$ This is the single equation in \(x\) and \(y\) that the exercise is asking us to find.

Step 2: Describe the curve

To describe the curve, we need to identify the type of curve that the equation \(x^2 + y = 1\) represents. This equation can be rearranged into the form of a standard equation for an ellipse with horizontal major axis: $$\frac{x^2}{1} + \frac{y}{1} = 1$$ However, since the range of \(t\) is from \(0\) to \(\pi\), we are only considering a portion of the ellipse. The curve consists of only the upper half of the unit circle, from \((1,0)\) to \((-1,0)\). The positive orientation of the curve means that as \(t\) increases, the curve moves in the counterclockwise direction. To summarize the solution: a. The single equation in \(x\) and \(y\) that represents the given parametric equations is \(x^2 + y = 1\). b. The curve represented by these parametric equations is the upper half of a unit circle, and its positive orientation is counterclockwise.

Key Concepts

Eliminating the Parameter

In the world of parametric equations, we often encounter the need to translate complex, motion-related expressions into a more familiar Cartesian form. Eliminating the parameter is akin to decoding a message, where we transform the given data into an intelligible format that unveils the real path on the graph.

Let's take our example, where we have the parametric equations \(x = \cos t\) and \(y = \sin^2 t\). Our mission is to find a connection exclusively between \(x\) and \(y\), bypassing the parameter \(t\). Crucial to this process is the application of trigonometric identities, in this case, the Pythagorean identity which asserts \(\sin^2 t + \cos^2 t = 1\). By rewriting \(x\) as \(\cos t\) and then squaring both sides, we obtain \(x^2 = \cos^2 t\) which seamlessly plugs into the identity to exclude \(t\) and gives us \(x^2 + y = 1\), a pristine Cartesian equation.

Trigonometric Identities

Like the multipurpose tools in a Swiss Army knife, trigonometric identities are versatile instruments in mathematics, adept at simplifying equations and cracking open complex problems. These identities are not just equations but pivotal relations that hold the universe of trigonometry together.

In our exercise, we make use of the fundamental Pythagorean identity to interlink \(x\) and \(y\). This identity is a cornerstone in trigonometry, forming the basis for more complex relationships and aiding in the transition from parametric to Cartesian equations. Knowing when and how to apply these identities is not just helpful, it is essential in untangling the knots posed by parametric descriptions and revealing the curves they describe.

Describe the Curve

Describing the curve of parametric equations is an exploratory journey into the shape that a mathematical function takes in a plane. Think of it as drawing a picture based solely on instructions given by the equations. The equation \(x^2 + y = 1\), derived from our parametric expressions, paints the image of a circle's upper hemisphere.

Navigating the Curve

Imagine that you're steering a boat with coordinates \(x\) and \(y\) through a river. These coordinated moves outline the curve of your path. In our case, the constraint \(0 \leq t \leq \pi\) steers our vessel only through the upper arc of the full circular path, from \(1,0\) to \(\-1,0\), thus delineating the top half of a unit circle on the Cartesian grid.

Positive Orientation of a Curve

The positive orientation of a curve is like the flow of traffic in a one-way street designated by a set of parametric equations. It defines the direction you would trace the curve, starting at one point and moving towards another, according to the increasing values of the parameter.

In our task, as \(t\) ascends from 0 to \(\pi\), our curve traces a counterclockwise trajectory along the upper half of the unit circle. This mirrors the default motion of positive orientation, akin to the hands of a clock moving 'backwards'. It exemplifies a harmonious agreement with many natural and mathematical phenomena that adopt a similar convention for defining positive rotation or progression.