# Astronomy - Latitude and Longitude problem

1. Oct 22, 2007

### Pomico

1. The problem statement, all variables and given/known data

An airliner flies from London Heathrow (latitude 51$$^{o}$$30' N, longitude 0$$^{o}$$ 10' W) to Tokyo Narita (latitude 35$$^{o}$$ 40' N, longitude 139$$^{o}$$ 45' E). Assuming that the Earth is spherical and the aeroplane takes a great circle route,

a)what is the total distance flown in nautical miles
b)in what direction (bearing E of N) does the pilot set out from Heathrow
c)what is the most northerly location (latitude and longitude) that the plane reaches on its great circle route?

2. Relevant equations

sinA/sina = sinB/sinb = sinC/sinc
cosa=cosb.cosc+sinb.sinc.cosA

3. The attempt at a solution

I have completed parts a) and b) and have found the answers to be 5161 nautical miles and 31.6$$^{o}$$ respectively.

For part c I have found the latitude to be 70.95$$^{o}$$ but can't think how to find the longitude as the spherical triangle I have drawn doesn't have enough information and I don't know how to go about finding what I need in order to get the answer.
I have attached a copy of my spherical triangle but it's rather small, my artistic skills are rather limited using a laptop with no mouse...
On my diagram point N is the North Pole, point H is Heathrow and point P is the most northerly point. All of the numerical values given are in degrees. I have another diagram including point T for Tokyo but have omitted this as I don't think it's necessary for the task. If it is please say, and I will upload it.

2. Oct 22, 2007

### Pomico

One thing I forgot, on my diagram the distance d is the longitude of P plus the longitude of H.
I assumed I would go about the second part of the problem by finding d and subtracting the longitude of H as that is essentially what I did for the first part in finding the latitude of P.

3. Oct 24, 2007

### Pomico

Never mind, solved it now!

4. Oct 26, 2007

### jim mcnamara

Consider posting your solution so that others can search the forums and find what you did.