Answer:
C = 45.8 or C = 117.69
Explanation:
Remark
Only SSA gives the possibility of 2 answers. This one does not give that opportunity. There is one unique answer. We'll discuss 2 and zero after finding 1 answer. On looking at it again, the question might be ambiguous. We'll check that out as well.
Given
AB = 15
<A = 35
point C opposite line AB such that CB = 12 These givens give a unique answer.
Solution
Sin(35) / 12 = Sin(C) / 15 Multiply both sides by 15
15*Sin(35) / 12 = Sin(C) Find 15/12
1.25*sin(35) = Sin(C) Write Sin(35)
1.25*0.5736 = Sin(C) Multiply the left
0.71697 = Sin(C) Take the inverse Sin
C = 45.805 degrees This is the angle opposite AB
Angle B = 180 - 35 - 45.805 = 99.2
Ambiguous Case
If AC = 12 we have another answer entirely. This is SAS which will give just 1 set of answers for the triangle. The reason the case is ambiguous is because we don't exactly know where that 12 unit line is. It could be AC or BC.
I will set up the Sin law for you, and let you solve it
Sin(B) / 12 = Sin(35)/15
When you solve for Sin(B) as done above you, get 0.45886 from which B = 27.31 degrees
C = 180 - 35 - 27.31 = 117.69
So that's two values that C could have. I think that's all given these conditions.
Two Cases or None
<A = 35 degrees
AC = 15
CB = 12
This should give you two possible cases or none. You can check which by finding the height of the triangle from C down to AB (which has no distinct length. The h is 15 * Sin35 = 8.6. If CB < 8.6, there are no solutions. If CB < AC then if CB > that 8.6, there are 2 solutions.