SOLUTION: 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 triang

Algebra ->  Trigonometry-basics -> SOLUTION: 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 triang      Log On


   



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) About Me  (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) About Me  (Show Source):
You can put this solution on YOUR website!



given:
alpha+=+111°
a+=+12
b+=+23
if triangle, from given data we know that alpha+=+111° is the largest angle, and opposite side a must be largest side which is not in your case
Since the side a%3Cb, for ASS+(SSA) theorem with A+%3E=+90 (A+%3E=+pi%2F2) and a+%3C=+b, there are no solutions and no+triangle!
answer: IMPOSSIBLE

Answer by math_tutor2020(3817) About Me  (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 -1+%3C=+sin%28x%29+%3C=+1
The largest sine can get is 1.

Answer: impossible