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Chapter 11: Problem 21

Graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. $$ x(t)=2 \cos t, \quad y(t)=3 \sin t ; \quad-\pi \leq t \leq 0 $$

Short Answer

Expert verified
The curve is an ellipse given by \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \), oriented from right to left as \( t \) goes from \( 0 \) to \( -\pi \).

Step by step solution

01

Identify the Parametric Equations

The given parametric equations are: \[ x(t) = 2 \cos(t), \quad y(t) = 3 \sin(t) \]. The parameter \( t \) ranges from \( -\pi \) to \( 0 \).
02

Sketch the Parametric Curve

To sketch the parametric curve, generate points by choosing values of \( t \) within the given range and compute the corresponding \( x \) and \( y \) coordinates. A table of values can be helpful here:t | \( x(t) \) | \( y(t) \) ---|-----------|----------- -\( \pi \) | 2\(-1\) | 3\(0\) -\( \pi/2 \) | 2\(0\) | 3\(-1\) 0 | 2\(1\) | 3\(0\) Plot these points and sketch a smooth curve through them. The orientation of the curve follows the direction of the increasing t values.
03

Find the Rectangular Equation

To find the rectangular equation, eliminate the parameter \( t \). Start with the parametric equations: \[ x = 2 \cos(t) \quad \text{and} \quad y = 3 \sin(t) \]. Isolate \( \cos(t) \) and \( \sin(t) \): \[ \cos(t) = \frac{x}{2}, \quad \sin(t) = \frac{y}{3} \]. Use the Pythagorean identity \( \cos^{2}(t) + \sin^{2}(t) = 1 \): \[ \left( \frac{x}{2} \right)^{2} + \left( \frac{y}{3} \right)^{2} = 1 \]. The rectangular equation of the curve is: \[ \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Curves
Parametric curves offer a unique way to describe geometric shapes. Unlike regular equations that relate x and y directly, parametric equations use a third variable, often called the parameter, denoted as t. This parameter allows us to express both x and y as functions of t. For example, in our exercise, the parametric equations are given by \( x(t) = 2 \cos(t) \) and \( y(t) = 3 \sin(t) \). By varying the parameter t within a specified range, we can generate a series of points (x, y) that trace out the curve. This approach is particularly powerful because it can represent more complex shapes, including those that are difficult to describe using traditional rectangular equations.
Rectangular Equations
Rectangular equations are the more familiar forms of equations that relate x and y directly, like \( y = mx + b \) or \( x^2 + y^2 = r^2 \). These types of equations are often simpler to graph and understand but may not always capture the full complexity of a curve. To convert from a parametric form to a rectangular form, we eliminate the parameter t by using trigonometric identities or algebraic manipulation. In our exercise, we started with the parametric equations \( x = 2 \cos(t) \) and \( y = 3 \sin(t) \) and aimed to find an equation that only involves x and y.
Parametric to Rectangular Conversion
Converting parametric equations to a rectangular equation involves a few algebraic steps. First, express the trigonometric functions in terms of x and y. For \( x = 2 \cos(t) \) and \( y = 3 \sin(t) \), we isolate \( \cos(t) \) and \( \sin(t) \): \( \cos(t) = \frac{x}{2} \) and \( \sin(t) = \frac{y}{3} \). Next, we use the Pythagorean identity \( \cos^2(t) + \sin^2(t) = 1 \): \[ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{3} \right)^2 = 1 \]. Simplify this to obtain the rectangular equation: \[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]. Now we have a rectangular form, which represents an ellipse.
Graphing Parametric Equations
Graphing parametric equations involves plotting points calculated from the parameter t, and then connecting the dots smoothly. In our exercise, we varied t from \( -\pi \) to 0. We calculated points: \( t = -\pi \), \( (x, y) = (-2, 0) \); \( t = -\frac{\pi}{2} \), \( (x, y) = (0, -3) \); and \( t = 0 \), \( (x, y) = (2, 0) \). Plot these points and notice the smooth curve and its orientation. The orientation shows the direction of increasing t, which helps to understand the path traced by the curve. This method gives a more intuitive grasp of the path's shape and direction over time, enhancing our understanding of the parametric relationship.

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