Find the distance between two points in polar coordinates (r1, θ1) and (r2, θ2) using the law of cosines.

• A. d = sqrt((r1 − r2)^2 + 4 r1 r2 sin^2((θ1 − θ2)/2))
• B. d = sqrt(r1^2 + r2^2 − 2 r1 r2 cos(θ1 − θ2))
• C. d = sqrt(r1^2 + r2^2 + 2 r1 r2 cos(θ1 + θ2))
• D. d = sqrt(r1^2 + r2^2 − 2 r1 r2 sin(θ1 − θ2))

Answer: (B)

Think of the two polar points as vectors from the origin: one has length r1 and angle θ1, the other length r2 and angle θ2. The distance between their endpoints forms a triangle with sides r1, r2, and the distance d between the points, and the angle between the two radius vectors is the difference Δ = θ1 − θ2. By the law of cosines, the distance satisfies

d^2 = r1^2 + r2^2 − 2 r1 r2 cos(Δ).

So the distance is d = sqrt(r1^2 + r2^2 − 2 r1 r2 cos(θ1 − θ2)). This is the standard form in polar coordinates.

You can also rewrite it using the identity cos(Δ) = 1 − 2 sin^2(Δ/2), giving

d^2 = (r1 − r2)^2 + 4 r1 r2 sin^2((θ1 − θ2)/2),

which is the same distance in a different but equivalent form. The key point is that the cosine of the difference of the angles governs the separation, not the sum.