   Chapter 10.2, Problem 33E

Chapter
Section
Textbook Problem

# Comparing Plane Curves In Exercises 31–34, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain.(a) x = cos   θ y = 2   sin 2 θ 0 < θ < π x = cos ( - θ ) y = 2   sin 2 ( − θ ) 0 < θ < π

To determine

Any difference between the curves of parametric equations,x=cosθ,y=2sin2θ and x=cos(θ),y=2sin2(θ) in the interval 0<θ<π. Also, check if the graphs, orientations and the smoothness of the curves are same or not.

Explanation

Given:

(a) The parametric equation, x=cosθ,y=2sin2θ.

(b) The parametric equation, x=cos(θ),y=2sin2(θ).

Consider the parametric equation,

x=cosθy=2sin2θ

Trigonometric formula gives,

sin2θ+cos2θ=1sin2θ=1cos2θ

Substitute the value sin2θ=1cos2θ in the equation y=2sin2θ,

y=2(1cos2θ)

Since, x=cosθ. Therefore, the above equation will be,

y=2(1x2)y=22x2

The equation y=22x2 is a parabola.

To draw the graph, assume different value of θ where 0<θ<π and find corresponding values of x and y,

For θ=0,

x=cosθ=cos(0)=1

And,

y=2sin2θ=2sin2(0)=2×(0)2=0

For θ=π4,

x=cosθ=cos(π4)=12=0.71

And,

y=2sin2θ=2sin2(π4)=2×(12)2=1

For θ=π3,

x=cosθ=cos(π3)=12=0.5

And,

y=2sin2θ=2sin2(π3)=2×(32)2=1

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Exercises 7-12, solve for y in terms of x. y24x2=2

Calculus: An Applied Approach (MindTap Course List)

#### Multiply: (12)(6)

Elementary Technical Mathematics

#### If cot=2120, find csc and sin

Elementary Geometry For College Students, 7e

#### For , f′(x) =

Study Guide for Stewart's Multivariable Calculus, 8th

#### True or False: The function y = f(x) graphed at the right could be a solution to y = x + y.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 