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Chapter 10: Problem 19

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=\cos \theta, y=-2 \sin ^{2} 2 \theta ;-\infty<\theta<\infty $$

Short Answer

Expert verified
(a) Plot the parametric form for visualization. (b) The curve relates through periodic trigonometric functions, suggesting non-simple and closed properties. (c) The Cartesian equation is \( y = -16x^4 + 16x^2 - 2 \).

Step by step solution

01

Understanding the Parametric Equations

We are given the parametric equations: \( x = \cos \theta \) and \( y = -2 \sin^2 2\theta \). Here, \( \theta \) is the parameter that describes the curve, and it ranges from \(-\infty\) to \(\infty\).
02

Graphing the Curve

To graph the curve, we explore the values that \( x \) and \( y \) take. The range for \( x \) is from -1 to 1 since cosine varies between -1 and 1. We calculate a few points by selecting values of \( \theta \) and plotting corresponding \( (x, y) \) values. This will help us visualize the curve. Use a graphing software or calculator if necessary to plot these points.
03

Investigating Curve Properties

A curve is simple if it doesn't intersect itself, and it's closed if it forms a loop where the endpoint coincides with the start point. For \(-\infty < \theta < \infty\), we check whether \( x \) and \( y \) ever return to the same values as \( \theta \) varies, which would suggest closedness. Using trigonometric identities, observe periodic patterns of sine and cosine that may indicate such properties.
04

Eliminating the Parameter

To eliminate the parameter and obtain a Cartesian equation, we use trigonometric identities. Start with \( x = \cos \theta \), which implies \( \cos \theta = x \). Then, substitute in \( y = -2 \sin^2 2\theta \). Recall, \( \sin^2 2\theta = \frac{1 - \cos 4\theta}{2} \). We need to express \( \cos 4\theta \) in terms of \( x \). Use the double angle identities: \( \cos 2\theta = 2\cos^2 \theta - 1 = 2x^2 - 1 \), and \( \cos 4\theta = 2\cos^2 2\theta - 1 = 2(2x^2 - 1)^2 - 1 \). Substitute back to find \( y \) in terms of \( x \).
05

Formulating the Equation

Simplify the relation \( y = -2\sin^2 2\theta = -\left(1 - \cos 4\theta \right)\) and substitute \( \cos 4\theta = 2(2x^2 - 1)^2 - 1 \) to get the equation: \( y = -\left(1 - (2(2x^2 - 1)^2 - 1) \right) \) which simplifies to: \[ y = -2 (8x^4 - 8x^2 + 1) = -16x^4 + 16x^2 - 2 \]
06

Checking the Final Form

Ensure that all steps in converting the parametric equations involve correct trigonometric identities and algebraic manipulations. Double check substitutions and ensure they lead to the consistent Cartesian equation. The final result integrates both parametric forms into a unified curve description.

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Key Concepts

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

Graphing Curves
Graphing parametric curves involves plotting points based on parametric equations. A parametric equation specifies the coordinates of the points on a curve as functions of an independent parameter. In our case, the parameter is \( \theta \) and the equations are \( x = \cos \theta \) and \( y = -2 \sin^2 2\theta \).
As \( \theta \) changes, \( x \) varies between -1 and 1 because the cosine function ranges from -1 to 1.
  • Begin by selecting a range of \( \theta \) values.
  • Calculate \( x \) and \( y \) for each \( \theta \).
  • Connect these points smoothly, as they form the continuous curve.
For practical graphing, it's often helpful to use graphing tools to visualize the complex shapes these curves can take.
Trigonometric Identities
Trigonometric identities are formulas that express relationships between the angles and lengths of triangles. They are especially useful in simplifying expressions involving trigonometric functions. In the context of our parametric equations, we used several key identities.The double angle identities allow us to relate \( \sin^2 \theta \) and \( \cos^2 \theta \) to other angles. For instance:
  • \( \cos 2\theta = 2\cos^2 \theta - 1 \)
  • \( \sin^2 2\theta = \frac{1 - \cos 4\theta}{2} \)
These identities help transform our parametric equations into a more manageable form for finding a Cartesian equation. They reveal patterns inherent in periodic functions, such as sine and cosine.
Cartesian Equations
A Cartesian equation represents a curve by relating \( x \) and \( y \) in a single equation. To convert parametric equations into a Cartesian format, we eliminate the parameter by expressing \( y \) solely in terms of \( x \).
Here's how we proceed:
  • Start from \( x = \cos \theta \), implying that \( \theta = \arccos x \).
  • Using \( y = -2 \sin^2 2\theta \), substitute and simplify using identities such as \( \cos 4\theta \).
  • Eventually, obtain a polynomial or other expression in terms of \( x \).
The algebra involved often demands careful manipulation and substitution of trigonometric identities to seamlessly transition from parametric to Cartesian form.
Curve Properties
Analyzing curve properties involves understanding certain geometric features, such as whether a curve is closed or simple. A closed curve loops back onto itself, while a simple curve does not intersect itself.
For the given parametric equations, curve properties can be inspected using periodicity:
  • Sine and cosine are periodic, which implies repetitiveness in the path of the curve.
  • Calculate intersection points by checking if and when \( (x, y) \) pairs repeat over specified intervals of \( \theta \).
By investigating these features, you can determine the nature of the curve's path and decide if it meets criteria for being closed or simple.

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Most popular questions from this chapter

In Problems 31-34, find the equation of the tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=3 t, y=8 t^{3} ; t=-\frac{1}{2} $$

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=9 \sin ^{2} \theta, y=9 \cos ^{2} \theta ; 0 \leq \theta \leq \pi $$

In Problems 31-34, find the equation of the tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=2 e^{t}, y=\frac{1}{3} e^{-t} ; t=0 $$

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3 \tan t-1, y=5 \sec t+2 ; t \neq \frac{(2 n+1) \pi}{2} $$

Plot the points whose polar coordinates are \((3,2 \pi)\), \(\left(2, \frac{1}{2} \pi\right), \quad\left(4,-\frac{1}{3} \pi\right), \quad(0,0),(1,54 \pi), \quad\left(3,-\frac{1}{6} \pi\right), \quad\left(1, \frac{1}{2} \pi\right), \quad\) and \(\left(3,-\frac{3}{2} \pi\right)\).
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