   Chapter 10.4, Problem 72E

Chapter
Section
Textbook Problem

Horizontal Tangency In Exercises 71 and 72, find the points of horizontal tangency to the polar curve. r = a sin θ cos 2 θ

To determine

To calculate: The points of horizontal tangency to the polar curve given as r=asinθcos2θ.

Explanation

Given:

The polar equation r=asinθcos2θ.

Formula used:

Product rule of differentiation ddx(f(x)g(x))=g(x)df(x)d(x)+f(x)dg(x)d(x).

Use the formula: cos2θ=cos2θsin2θ

Calculation:

The polar equation is,

r=asinθcos2θ …… (1)

Convert the polar equation into rectangular coordinates.

x=rcosθ=(asinθcos2θ)cosθ=asinθcos3θ

And,

y=rsinθ=(asinθcos2θ)sinθ=asin2θcos2θ

Differentiate the equation given as, y=asin2θcos2θ with respect to θ is,

dydθ=ddθ(asin2θcos2θ)=ddθ(sin2θ)(acos2θ)+(asin2θ)ddθ(cos2θ)=(acos2θ)2sinθcosθ(asin2θ)2cosθsinθ

Simplified further, to get;

dydθ=

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