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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the curves r = a sin θ and r = a cos θ intersect at right angles.

To determine

To show: The curves r=asinθ and r=acosθ intersect at right angles.

Explanation

Given:

The equations of curves are r=asinθ and r=acosθ .

Calculation:

The curve r=asinθ with horizontal tangent is shown in below figure 1.

Write the slope formula for curves.

dydx=dydθsinθ+rcosθdxdθcosθrsinθ

For the curve r=asinθ , (m1) be the slope of the tangent.

Let (dydx) be m1 .

Differentiate the curve equation r with respect to θ .

drdθ=acosθ

m1=drdθsinθ+rcosθdrdθcosθrsinθ

Substitute (acosθ) for drdθ in equation drdθsinθ+rcosθdrdθcosθrsinθ

dydx=(acosθ)sinθ+rcosθ(acosθ)cosθrsinθ=2cosθsinθcos2θsin2θdydx=sin2θcos2θm1=sin2θcos2θ

The curve r=acosθ with vertical tangent is shown in below figure 2.

Write the slope formula for curves.

dydx=dydθsinθ+rcosθdxdθcosθrsinθ

For the curve r=acosθ , (m2) be the slope of the tangent

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