This is considering the earth as a sphere with a radius of 3960 miles.
What is did was to first find the circumference: $2\pi r=2\pi (3960)=7920\pi$.
Then, since the points differ by $d$ degrees in their longitudes, I multiplied the circumference by $d/360$, since I only want $d/360$'s worth of the circumference of the great circle, so I got $22\pi d$.
I have a feeling that I used $d$ wrong, and that maybe I should consider the latitude or the points; or am I heading in the right direction?
I was asked to find the distance of the two points using the circumference of the circle that contains that two points and whose points are all on the same latitude and, using a similar method as above, I got that the distance was $22\pi d \cos (L)$. I had to show that the distance from the great circle was smaller than the distance from this latitude circle, but that isn't the case in my calculations. Can anyone find my mistake?
Best Answer
In the picture below, $R=3960$ miles is the Earth radius, $r=R\cos L$ (where $L$ is the latitude) and angle $d$ represents the difference in longitude between points $A$ and $B$ ($d\le180°$). The circle containing the two points and whose points are all at the same latitude is the orange circle in diagram below: the length of arc $AB$ along that circle is ${\pi\over180°}d r={\pi\over180°}d R\cos L$ (I'm using degrees for the angles).
The arc of great circle passing through $A$ and $B$ has center $O$, thus its length is ${\pi\over180°}\theta R$. To find $\theta$, we can compute the rectilinear distance $AB$ in two ways: $$ AB = 2r\sin{d\over2}=2R\sin{\theta\over2}, \quad\hbox{whence:}\quad \sin{\theta\over2}=\sin{d\over2}\cos L. $$ The length of great arc $AB$ is then $$ 2R{\pi\over180°}\arcsin\left(\sin{d\over2}\cos L\right). $$