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Math Help - Law of sines problem ?

  1. #1
    Junior Member xterminal01's Avatar
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    Law of sines problem ?

    Two fire-lookout stations are 10 miles apart, with station B directly east of station A. Both stations spot a fire. The bearing of the fire from station A is N25E and the bearing of the fire from station B is N56W. How far, to the nearest tenth of a mile, is the fire from each lookout station?


    So I got side a = 4.28 b=8.4 and angle C 180 - (25+56) = 99
    But the books says the station A is 6 miles and station B about 9 miles from fire?


    Where exactly did i mess up?
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  2. #2
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    You've gone wrong at the very beginning. This is hard to explain without a diagram but I'll try. If you form a diagram where A and B are the corners of the base and that length is obviously 10. Draw a line straight up from A (know as your north line) it should be at a right angle to the base of your triangle. You've been told the fire is N25E so draw a line 25 degrees clockwise from you're north line. Call the point where the fire is F. Put in a north line from B as well. You are also told the fire is N56W of B meaning you draw a line 56 degrees anti-clockwise from B to the point F.

    You have used the angles given in you formula when it should be the angles which are inside the triangle you have formed.

    So the angle at A should be 90-25 = 65 and similarly with B it should be 90-56 = 34. Everything should now come out with the same answer.

    Again as I said it's not easy without a diagram so ask if you didn't understand that .
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  3. #3
    Junior Member xterminal01's Avatar
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    Thanks,

    I got the correct result; however, I still dont understand why A is 65 and B = 34. Where does the 90-25 = 65 come from?
    I got as angel F where the two meet 81 and i got the sides correct.
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  4. #4
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    As you should know the sine rule is used with triangles. The angle you used originally was the angle formed between the north line and the outside of the line which formed the triangle.

    In the case of B the angle between the north line and the line BF is 56 degrees because bearings are measured from north and the instruction N56W means 56 degrees in the north west direction (anti - clockwise). However to apply the sine rule you need the angle within the triangle AFB. You know the whole angle between the north line and the line AB is 90 degrees and subtracting the angle outside the triangle (56 degrees) give the angle inside the triangle.

    With the case is the same for the angle at A inside the triangle except when measuring the bearing the instruction N25E means measure 35 degrees in the north east direction from the north line.

    I've attached an image which will hopefully work and provide a better understanding. Sorry for the bad drawing i'm not the best at art or drawing in paint.
    Attached Thumbnails Attached Thumbnails Law of sines problem ?-bearings.jpg  
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  5. #5
    Junior Member xterminal01's Avatar
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    Thanks Amanda,
    Now I got it, the drawing illustrated exactly what you were saying.
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