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Math Help - Finding approximate latitude/longitude to the nearest minute (trigonometric)

  1. #1
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    Finding approximate latitude/longitude to the nearest minute (trigonometric)

    Hello, this is my first post here on mathhelpforum. I hope to become an active member, helping and posting whatever I can/know to contribute.

    I have been working on a problem that gives me three locations: A B and C. I have the latitude and longitude of A and B (approximate), and I have the degrees in which the directions of A B and C are pointing. By using a latitude and longitude to miles converter, I was able to find the length of side c, and since I had an opposite angle, I was able to use the Law of Sines to find the other sides, thus forming a triangle with all sides abc and angles ABC.

    So I have the latitude and longitude of A (N46, W115.5), and the latitude and longitude of B (N48.16, W118.75).

    A=35.83 degrees
    B=54.16 degrees
    C=90.01 degrees
    a=125.20 miles
    b=173.83 miles
    c=213.87 miles

    Given the information, what is the best way to approach finding the latitude and longitude of C to the nearest minute? I was guessing I could make a coordinate system, treating west as X and north as Y... I have those plotted... and do not know where to go from there.

    Any help pushing me towards the right direction would be greatly appreciated.
    See you around!
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  2. #2
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    There's one important thing to remember.

    1 longitude is 1 longitude, no matter where you are.

    1 lattitude is substantially larger at the equator than 1 near the North or South Pole.

    Generally, a good approximation can be achieved by attaching sin(longitude) to your lattitudes.

    Let's see what you get.
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  3. #3
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    Alright, so since I have the distances between abc, and since longitude is longitude no matter what, I began by using the distance between segment AB as a meter the find the miles / degree.

    B = 118.75 long
    A = 115.5 long

    B - A = 3.25 difference.
    c/3.25 = 65.80 miles/degree.

    So since I have the distance between segment AC, I used the unit conversion with b to find the degree difference, and then subtract this difference from A's longitude?

    b/65.80 = 2.64
    115.5-2.64 = 112.86

    Convert to the nearest second... 112 51' 36"

    Have I found the longitude of C correctly? ... or am I going about this the wrong way?
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  4. #4
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    You will have to worry about geographical topography, but that is the idea. Try a few known cities and check it out.
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  5. #5
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    Quote Originally Posted by TKHunny View Post
    There's one important thing to remember.

    1 longitude is 1 longitude, no matter where you are.

    1 lattitude is substantially larger at the equator than 1 near the North or South Pole.
    TKHunny, I believe you have those reversed. Longitude goes east to west, latitude north and south. 1 of longitude, near the equator, is 1/360 of the circumference of the earth, about 21600/360= 60 nautical miles (in fact, the nautical mile is defined to be the length of 1 minute in longitude at the equator- it is remarkable that a nautical mile is so close to a statute mile). While close to the poles, the length of 1 longitude goes to 0.

    Jeffh, what you really need is "spherical trigonometry" which used to be taught at the secondary level but is barely mentioned now.

    This might help: http://www.boeing-727.com/Data/fly%20odds/distance.html

    Generally, a good approximation can be achieved by attaching sin(longitude) to your lattitudes.

    Let's see what you get.
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  6. #6
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    I am always doing that. Not sure why it won't stick. Sorry for reversal.

    Note: If you see me in the cockpit of your airliner, please check to see who is navigating!
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