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. $$x=\cos t, y=\sin ^{2} t ; 0 \leq t \leq \pi$$

Short answer

Answer: The Cartesian equation of the curve is \(y = 1 - x^2\). It represents a parabola opening downwards, with the vertex at (0, 1). The positive orientation is moving from right to left along the curve as \(t\) increases from \(0\) to \(\pi\).

Step-by-step solution

In the given parametric equation \(x = \cos t\), we can simply take the inverse cosine on both sides to get \(t = \arccos{x}\). #Step 2: Substitute the expression of t into the second equation#

Using the expression of \(t\) in terms of \(x\) from step 1, we can now substitute it into the second equation: \(y = \sin^2 t\). So, we now have: \(y = \sin^2 (\arccos x)\). #Step 3: Simplify the equation to get the Cartesian equation#

We can use the Pythagorean identity to replace \(\sin^2 t\) with \(1-\cos^2 t\) since \(\sin^2 t + \cos^2 t = 1\). Therefore, we get: \(y = 1 - \cos^2(\arccos x)\). Since \(\cos(\arccos x) = x\), we can now write the final equation as: $$y = 1 - x^2$$ #Step 4: Find the curve's orientation using the given range of t#

The given range of \(t\) is \(0 \leq t \leq \pi\). When \(t = 0\), we have \(x = \cos 0 = 1\) and \(y = \sin^2 0 = 0\). As \(t\) increases from \(0\) to \(\pi\), we can see that \(x\) decreases (as \(\cos t\) decreases) and \(y\) increases (as \(\sin^2 t\) increases). So, the positive orientation is moving from right to left along the curve, as t increases from \(0\) to \(\pi\). To summarize the solution: 1. The Cartesian equation of the curve is \(y = 1 - x^2\). 2. It represents a parabola opening downwards, with the vertex at \((0, 1)\). 3. The positive orientation is moving from the right to the left along the curve.

Key concepts

Cartesian Equation

The Cartesian equation is a way of expressing a mathematical equation using coordinates based on the Cartesian coordinate system, where each point is defined by an \(x\) and a \(y\) coordinate. In this exercise, we start with parametric equations, which express \(x\) and \(y\) in terms of a third variable, called the parameter (in this case, \(t\)). To eliminate the parameter and convert the parametric equations into a Cartesian equation, follow these steps:

1. Recognize the parametric equations: \(x = \cos t\) and \(y = \sin^2 t\).
2. We first express \(t\) in terms of \(x\) by isolating \(t\), which is done using the inverse cosine function: \(t = \arccos{x}\).
3. Substitute the \(t\) from the first equation into the second equation to find \(y\) in terms of \(x\): \(y = \sin^2(\arccos x)\).

This process transforms our parametric equations into a singular equation relating \(x\) and \(y\), which is known as the Cartesian equation. It simplifies our understanding of the relationship between \(x\) and \(y\) to \(y = 1 - x^2\), describing the shape of the curve in a more conventional and recognizable manner.

Curve Description

Describing the curve involves understanding both the shape and the behavior of its graph. The conversion from parametric to Cartesian revealed the equation \(y = 1 - x^2\), which represents a parabola. Here's how you can describe the curve comprehensively:

1. **Shape and Direction:** The curve is a parabola that opens downwards. This is indicated by the negative sign before \(x^2\).
2. **Vertex:** The vertex of this parabola is the highest point and it is located at \(x = 0\), \(y = 1\). This gives the vertex coordinates \((0, 1)\).
3. **Orientation:** The problem gives a range for \(t\), which affects the curve's orientation. From \(t = 0\) to \(t = \pi\), the \(x\) values decrease and \(y\) values correspondingly adjust, tracing the parabola from right to left.

In mathematical terms, this means as \(t\) changes, we graphically plot this parabola from the point at \( (1, 0) \) towards \((-1,0)\), following the specified interval \(0 \leq t \leq \pi\). This understanding aids in visualizing and sketching the curve accurately.

Pythagorean Identity

The Pythagorean identity is a critical tool in mathematics, especially when working with trigonometric functions. It states that \(\sin^2 t + \cos^2 t = 1\). This identity is particularly useful in this problem as it aids in eliminating the parameter \(t\).

1. **Understanding the identity:** It is derived from the geometry of a right-angled triangle and relates the squares of the sine and cosine of an angle, always summing to 1. For example, in our problem, we used this identity to express \(\sin^2 t\) in terms of \(\cos^2 t\): \(\sin^2 t = 1 - \cos^2 t\).
2. **Substitution:** By substituting \(\cos t = x\), where \(t\) can now be expressed in terms of \(x\), we simplified \(\sin^2(\arccos x)\) to \( 1 - x^2\).

This transformation simplifies the equations, allowing us to focus on the relationship between \(x\) and \(y\). The Pythagorean identity is thus a bridge to facilitate understanding how trigonometric functions can be simplified to Cartesian forms, removing complexities with ease.