   Chapter 1, Problem 44P

Chapter
Section
Textbook Problem

Given points (r1, θ1) and (r2, θ2) in polar coordinates, obtain a general formula for the distance between them. Simplify it as much as possible using the identity cos2 θ + sin2 θ = 1. Hint: Write the expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

To determine
The expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

Explanation

Given Info: The trigonometric identity is sin2θ+cos2θ=1 and (cosθ1cosθ2+sinθ1sinθ2)=cos(θ1θ2) .

Explanation:

Considering the below figure to get the Cartesian coordinates of the two points are,

x1=r1cosθ1and  y1=r1sinθ1

x2=r2cosθ2and  y2=r2sinθ2

The distance between the two points will be the hypotenuse of the shaded triangle which is given by,

(Δs)=(Δx)2+(Δy)2=(r12cos2θ1+r22cos2θ22r1r2cosθ1cosθ2)+(r12sin2θ1+r22sin2θ2

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