Solving Distance By Great Circle Sailing

Your ship departs Yokohama, Japan from position Lat. 35° 27' N; Long. 139° 39' E bound for San Francisco, California, USA. At position Lat. 37° 48.5' N, Long. 122° 24' W. Determine the distance by Great Circle Sailing.   

Given:

   Lat_{1 = 35}° 27' N

   Lat_{2 = 37}° 48.5' N

Long_{1 = 139}° 39' E

Long_{2 = 122}° 24' W

What is asked?

Distance by Great Circle Sailing

Solution:

1. Solve first for the difference of longitude (Dlo). Here's the formula in getting Dlo:

    If same name, subtract

    If different name, plus

    Note: Affix name of direction. If Dlo is greater than 180, subtract it to 360 and then affix name of  Long_1. 

                

               Long_{1 = 139}° 39' E

               Long_2 = 122° 24' W 

                            261° 63'

                           +   1°-60' 

                            262° 03'

                         - 360°        

                  Dlo =   97° 57' E

2. We can now proceed to solve for Great Circle distance. 

     Note: Plus (+) when not crossing the equator; Subtract (-) when crossing the equator.

          Cos GCD = (Cos Lat1 × Cos Lat2 × Cos Dlo) + (Sin Lat1 × Sin Lat2)    

        Cos GCD  = ( Cos ₃₅° 27' × Cos ₃₇° 48.5' × Cos 97° 57' ) + ( Sin ₃₅° 27' × Sin ₃₇° 48.5' ) 

          Cos GCD= -0.08902 + 0.35548

              GCD    = inv Cos 0.26646

                 GCD =  74.55°
                            ×   60        ( to convert to miles)

                 GCD = 4473 miles

           

Great Circle Sailing

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