Use the result in Exercise 1 to find a formula for the great circle distance between two p...

Use the result in Exercise 1 to find a formula for the great circle distance between two points on a sphere of radius R with coordinates (R, θ1, ϕ1) and (R, θ2, ϕ2). (See Exercise 2)

Exercise 1

Given two points P1 and P2 with spherical coordinates (ρ1, θ1, ϕ1) and (ρ2, θ2, ϕ2), show that

(Hint: You will need several trigonometric identities.)

Exercise 2

Given a sphere with center at the origin, the great circle through two points P1 and P2 on the sphere is the circle with center at the origin that passes through P1 and P2. The great circle distance between P1 and P2 is the length of the arc on the great circle. Find the great circle distance between the following pairs of points (given in spherical coordinates):

(a) (3, π/4, π/6), (3, π/4, 3π/4)

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(b) (2, π/3, π/2), (2, 7π/6, π/2)

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(c) (1, π/4, 0), (1, π/2, π/4)