document.write( "Question 1198037: Could you please help me with this problem about Law of Sines? I understand this is an Ambiguous case but I am getting an incorrect answer. Thank you in advance!\r
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Algebra.Com's Answer #831710 by MathTherapy(10556)\"\" \"About 
You can put this solution on YOUR website!
Could you please help me with this problem about Law of Sines? I understand this is an Ambiguous case but I am getting an incorrect answer. Thank you in advance!
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document.write( "I wonder what incorrect answer you're getting.\r\n" );
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document.write( "Let the 1st boat's location be D, the 2nd boat's, C, and yours, B\r\n" );
document.write( "Then DC = 800m = b, and BD = 1,000m = c\r\n" );
document.write( "We now apply the Law of Sines, \"matrix%281%2C3%2C+sin+%28B%29%2Fb%2C+%22=%22%2C+sin+%28C%29%2Fc%29\" to get: \"matrix%281%2C3%2C+sin+%2845%5Eo%29%2F800%2C+%22=%22%2C+sin+%28C%29%2F%221%2C000%22%29\"\r\n" );
document.write( "                                                   800 * sin (C) = 1,000 * sin (45o) ----- Cross-multiplying\r\n" );
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document.write( "With ∡s B and C being 45o and 62o, respectively, ∡D will be: 180 - (45 + 62) = 180 - 107 = 73o\r\n" );
document.write( "You need to know how far the 2nd boat is from you, which is BC or d\r\n" );
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document.write( "Again, apply the Law of Sines, \"matrix%281%2C3%2C+sin+%28D%29%2Fd%2C+%22=%22%2C+sin+%28B%29%2Fb%29\" to get: \"matrix%281%2C3%2C+sin+%2873%5Eo%29%2Fd%2C+%22=%22%2C+sin+%2845%5Eo%29%2F800%29\"\r\n" );
document.write( "                                                    d * sin (45o) = 800 * sin (73o) ----- Cross-multiplying\r\n" );
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document.write( "As seen, ∡C is the MIDDLE angle (62o), compared to 45o and 73o for ∡s B and D (the largest), respectively.\r\n" );
document.write( "However, ∡C can also be 118o (180o - 62o), thereby making it the largest angle, instead of D. \r\n" );
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document.write( "With ∡C being 118o, ∡B remaining at 45o (always will), ∡D will now be: 180 - (118 + 45) = 180 - 163 = 17o.\r\n" );
document.write( "This makes ∡D the smallest, which in turn makes d, or BC the shortest of the 3 sides, at 330.7808241 ≈ 330.78, or approximately 331 m.\r\n" );
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document.write( "So, unless you receive some clue (the 2nd boat is CLOSER to you than it is to the 1st boat, or the 2nd boat is FARTHER away from you \r\n" );
document.write( "than it is to the 1st boat) that'll indicate which of these 2 sides \"d\" could be, its TRUE distance from you cannot be determined\r\n" );
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document.write( "I forgot to check the 2 possibilities usng the TRIANGLE INEQUALITY THEOREM, hence the crossout above. The theorem states that ANY side\r\n" );
document.write( "of a triangle will be less than the sum of the triangle's other two sides.\r\n" );
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document.write( "Doing that for the 1st scenario (distance or length of 1,082), and using 1,082 as one of the triangle's sides, we determine if this is \r\n" );
document.write( "TRUE: 1,082 < 1,000 + 800. We clearly see that this is FALSE, so 1,082 m CANNOT be one of the sides of this triangle. Hence, such a triangle \r\n" );
document.write( "is NON-EXISTENT, and you CANNOT be 1,082 m from the 2nd boat. \r\n" );
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document.write( "We then try 331 m as one of the triangle's sides to determine if the following is TRUE: 331 < 1,000 + 800. We clearly see that this is\r\n" );
document.write( "TRUE, as 331 is definitely less than 1,000 + 800 = 1,800. \r\n" );
document.write( "Therefore, such a triangle DOES EXIST and so, the 2nd boat is approximately 331 m from you.
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