Friday, March 10, 2017

Solution to Spherical Trig Distance Problem

Problem:

Work out the distance from Barbados to Bristol using the spherical trig formula. (You can round off values to the nearest tenth.)


Solution:

We have for the relevant formula:

cos(dist.)= sin (lat1) sin (lat2) + cos (lat1) cos (lat2) cos (long1 - long2)


Where all '1' notations apply to Barbados, and all '2' notations apply to Bristol:

Then: lat1= 13 deg N     lat 2 = 51 deg 26' N =  51. 4 deg

long 1 = 59 deg 30'  W. = 59.5  deg,   long 2 =  2 deg 35' W = 2.7 deg

Where:

sin (lat1) =  sin (13)  =  0.225

sin (lat2) =sin (51.4) =    0.783

cos (lat 1) = cos(13)  = 0.974

cos (lat2) = cos (51.4) =   0.623


cos(long1 - long 2) = cos (59.5 - 2.7) = cos (56.8) =  0.548

Then:

cos(dist) = (0.225) (0.783) + (0.974)(0.623) (0.548)

cos (dist) =0.509

Then dist. = arc cos (0.593)  = 59.4 deg

Which is the great circle distance in degrees.

To convert to miles, we use the fact the spherical Earth has 360 degrees of longitude, and so:

24, 901 mi./   360 deg  =   69.1 miles per degree


Then multiplying this factor by the great circle distance (degrees):

D (miles) =   59.4 degrees (69.1 miles/ degree)

D (miles) =  4, 105  miles 

Which is confirmed by a Google check, i.e. "Distance from  Barbados to Bristol, UK"

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