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!
<pre>*[illustration ADC_Problem_1198037.png].
<font size = 3>I wonder what incorrect answer you're getting.

Let the 1st boat's location be D, the 2nd boat's, C, and yours, B
Then DC = 800m = b, and BD = 1,000m = c
We now apply the Law of Sines, {{{matrix(1,3, sin (B)/b, "=", sin (C)/c)}}} to get: {{{matrix(1,3, sin (45^o)/800, "=", sin (C)/"1,000")}}}
                                                   800 * sin (C) = 1,000 * sin (45<sup>o</sup>) ----- Cross-multiplying
                                                          {{{matrix(2,3, sin (C), "=", ("1,000" * sin (45^o))/800, sin (C), "=", (5 * sin (45^o))/4)}}}
                                                              {{{matrix(1,8, C, "=", sin^(- 1)((5 * sin(45^o))/4), "=", 62.11, "=", 62^o, "(approximately)"))}}}

With &#8737s B and C being 45<sup>o</sup> and 62<sup>o</sup>, respectively, &#8737D will be: 180 - (45 + 62) = 180 - 107 = 73<sup>o</sup>
You need to know how far the 2nd boat is from you, which is BC or d

Again, apply the Law of Sines, {{{matrix(1,3, sin (D)/d, "=", sin (B)/b)}}} to get: {{{matrix(1,3, sin (73^o)/d, "=", sin (45^o)/800)}}}
                                                    d * sin (45<sup>o</sup>) = 800 * sin (73<sup>o</sup>) ----- Cross-multiplying
                                                              {{{matrix(1,9, highlight(d), "=", (800 * sin (73^o))/sin (45^o), "=", "1,081.935325", "=", highlight("1,082"), m, "(approximately)"))}}}

As seen, &#8737C is the MIDDLE angle (62<sup>o</sup>), compared to 45<sup>o</sup> and 73<sup>o</sup> for &#8737s B and D (the largest), respectively.
However, &#8737C can also be 118<sup>o</sup> (180<sup>o</sup> - 62<sup>o</sup>), thereby making it the largest angle, instead of D. 

With &#8737C being 118<sup>o</sup>, &#8737B remaining at 45<sup>o</sup> (always will), &#8737D will now be: 180 - (118 + 45) = 180 - 163 = 17<b><sup>o</sup></b>.
This makes &#8737D the smallest, which in turn makes d, or BC the shortest of the 3 sides, at 330.7808241 ≈ 330.78, or approximately 331 m.

<s>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 
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</s>

I forgot to check the 2 possibilities usng the TRIANGLE INEQUALITY THEOREM, hence the crossout above. The theorem states that ANY side
of a triangle will be less than the sum of the triangle's other two sides.

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 
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 
is NON-EXISTENT, and you CANNOT be 1,082 m from the 2nd boat. 

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
TRUE, as 331 is definitely less than 1,000 + 800 = 1,800. 
Therefore, such a triangle DOES EXIST and so, the 2nd boat is approximately <font color = red><font size = 4><b>331 m</font></font></b> from you.</pre>