   Chapter 10.2, Problem 19E

Chapter
Section
Textbook Problem

# Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work.19. x = cos θ, y = cos 3θ

To determine

To find: The equation for horizontal or vertical tangent is horizontal lines and plot the curve for the parametric equations x=cosθ and y=cos3θ .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=cosθ (1)

The parametric equation for the variable y is as follows.

y=cos3θ (2)

Calculation:

Differentiate the parametric equation for x with respect to θ .

x=cosθdxdθ=sinθ

Differentiate the parametric equation for y with respect to θ .

y=cos3θdydt=3sin3θ

The tangent is horizontal when the expression dydt=0 is true.

dydt=03sin3θ=03θ=πθ=0,π,π3,2π3 (sinx=0x=nπ)

Substitute (0) for θ in equation (1).

x=cosθ=cos(0)=1

Substitute (0) for θ in equation (2).

y=cos3θ=cos3(0)=1

Substitute (π) for θ in equation (1).

x=cosθ=cos(π)=1

Substitute (π) for θ in equation (2).

y=cos3θ=cos3(π)=1

Substitute (π3) for θ in equation (1).

x=cosθ=cos(π3)=12

Substitute (π3) for θ in equation (2)

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