Astronomy - Latitude and Longitude problem

Pomico

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

An airliner flies from London Heathrow (latitude 51[tex]^{o}[/tex]30' N, longitude 0[tex]^{o}[/tex] 10' W) to Tokyo Narita (latitude 35[tex]^{o}[/tex] 40' N, longitude 139[tex]^{o}[/tex] 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[tex]^{o}[/tex] respectively.

For part c I have found the latitude to be 70.95[tex]^{o}[/tex] 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.

Thanks in advance!

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.

Pomico

Never mind, solved it now!

jim mcnamara

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