Chapter 10.4, Problem 72E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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=asinÎ¸cos2Î¸.

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: cos2Î¸=cos2Î¸âˆ’sin2Î¸

Calculation:

The polar equation is,

r=asinÎ¸cos2Î¸ â€¦â€¦ (1)

Convert the polar equation into rectangular coordinates.

x=rcosÎ¸=(asinÎ¸cos2Î¸)cosÎ¸=asinÎ¸cos3Î¸

And,

y=rsinÎ¸=(asinÎ¸cos2Î¸)sinÎ¸=asin2Î¸cos2Î¸

Differentiate the equation given as, y=asin2Î¸cos2Î¸ with respect to Î¸ is,

dydÎ¸=ddÎ¸(asin2Î¸cos2Î¸)=ddÎ¸(sin2Î¸)(acos2Î¸)+(asin2Î¸)ddÎ¸(cos2Î¸)=(acos2Î¸)2sinÎ¸cosÎ¸âˆ’(asin2Î¸)2cosÎ¸sinÎ¸

Simplified further, to get;

dydÎ¸=

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