   Chapter 10.3, Problem 14E

Chapter
Section
Textbook Problem

Find a formula for the distance between the points with polar coordinates (r1, θ1) and (r2, θ2).

To determine

To find: The distance formula between the points with the polar coordinates (r1,θ) and (r2,θ).

Explanation

Calculation:

The Cartesian coordinates for the variable x1 is as below.

x1=r1cosθ1

The Cartesian equation for the variable y1 is as below.

y1=r1sinθ1

The Cartesian coordinates for the variable x2 is as below.

x2=r2cosθ2

The Cartesian equation for the variable y2 is as below.

y2=r2sinθ2

The distance formula is as given below.

d=(x2x1)2+(y2y1)2=(r2cosθ2r1cosθ1)2+(r2cosθ2r1cosθ1)2=r22cos2θ2+r12cos2θ12r1r2cosθ1cosθ2+r22sin2θ2+r12sin2θ12r1r2sinθ

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