Question 1204634: Assume 𝛼 is opposite side a, 𝛽 is opposite side b, and 𝛾 is opposite side c. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth. (If not possible, enter IMPOSSIBLE.)
𝛼 = 111°,
a = 12,
b = 23
Found 3 solutions by Edwin McCravy, MathLover1, math_tutor2020: Answer by Edwin McCravy(20060) (Show Source):
You can put this solution on YOUR website!
It's impossible, because the longest side is always opposite the largest angle.
'α' is the largest angle because it's obtuse, and its opposite side, 'a', must
be the longest side. But 'b', at 23, is longer than 'a', at only 12.
Edwin
Answer by MathLover1(20850) (Show Source): Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
As mentioned by the other tutors, the triangle cannot be formed with these sides and given angle.
Using the argument "largest side opposite largest angle" is perhaps the most efficient pathway.
I'll show the law of sines scratch work that will lead to the same conclusion.
sin(alpha)/a = sin(beta)/b
sin(111)/12 = sin(beta)/23
sin(beta) = 23*sin(111)/12
sin(beta) = 1.789 approximately
Please make sure that your calculator is set to degrees mode.
We stop here since there are no real number solutions for angle beta.
Recall that 
The largest sine can get is 1.
Answer: impossible
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