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

Question 1205051: 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 = 117°,

a = 11,

b = 24
𝛽 =
°
𝛾 =
°
c =

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: Impossible

Work Shown

Law of Sines
sin(𝛼)/a = sin(𝛽)/b
sin(117)/11 = sin(𝛽)/24
sin(𝛽) = 24*sin(117)/11
sin(𝛽) = 1.94401 approximately

We stop here because the largest sin(𝛽) can get is 1.
sin(𝛽) = 1.94401 has no real number solutions.
We cannot solve this triangle simply because a triangle cannot exist with these sides and angle. This is the SSA case.

Here's another approach.
𝛼 = 117° is the largest angle. It is obtuse. Any obtuse triangle will have exactly one angle between 90 and 180 degrees excluding both endpoints.
The other two angles must be acute.

Since 𝛼 = 117° is the largest angle, the longest side must be side 'a'.
Longest side is opposite the largest angle.

However, b = 24 is larger than a = 11.
This contradiction is another way to see why this triangle is impossible

More practice with solving a triangle.
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204286.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204365.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204423.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204634.html

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