Question 1059132
i believe this has to do with the law of sines.


that law says that a / sin(A) = b / sin(B) = c / sin(C).


this can also be written as d / sin(D) = e / sin(E) = f / sin(F)


your triangle is DEF.


side opposite angle D is d.
side opposite angle E is e.
side opposite angle F is f.


you are given that angle D is equal to 66 degrees and side d is equal to 15 and side f is equal to 11.


by the law of sines, you get d / sin(D) = f / sin(F).


this becomes 15 / sin(66) = 11 / sin(F)


solve for sin(F) to get sin(F) = 11 * sin(66) / 15


solve for sin(F) to get sin(F) = .6699333356


solve for F to get F = arcsin(.6699333356) = 42.06191982 degrees.


since the sum of angle D and F is less than 180, then at least 1 triangle is possible.


to see if a second trianlge is possible, take 180 - 42.06191982 degrees to get 137.9380802 degrees.


add that to 66 degrees and the sum is greater than 180, so only 1 triangle is possible.


here's a reference that might help you to understand.


<a href = "http://www.regentsprep.org/regents/math/algtrig/att12/lawofsinesAmbiguous.htm" target = "_blank">http://www.regentsprep.org/regents/math/algtrig/att12/lawofsinesAmbiguous.htm</a>