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 1204423: 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.)
b = 3.8,
c = 5.9,
𝛾 = 80°
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if
b+=+3.8
c+=+5.9
gamma+=+80°

use the Law of Sines:

b%2Fsin%28beta%29=c%2Fsin%28gamma%29
3.8%2Fsin%28beta%29=5.9%2Fsin%2880%29
3.8sin%2880%29=5.9sin%28beta%29
sin%28beta%29=%283.8sin%2880%29%29%2F5.9
sin%28beta%29=0.6342829595671847
beta=sin%5E-1%280.6342829595671847%29
beta=39.4°

then
alpha=180-%28beta+%2B+gamma%29
alpha=180-%2839.4+%2B+80%29
alpha=60.6°


and side a

a%2Fsin%28alpha%29=c%2Fsin%28gamma%29
a%2Fsin%2860.63%29=5.9%2Fsin%2880%29
a=%285.9%2Fsin%2880%29%29sin%2860.63%29%29
a=5.22


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

Instead of alpha, beta, gamma, I'll use A,B,C.

Given info:
b = 3.8
c = 5.9
C = 80 degrees

Law of Sines
sin(B)/b = sin(C)/c
sin(B) = b*sin(C)/c
sin(B) = 3.8*sin(80)/5.9
sin(B) = 0.634283
B = arcsin(0.634283) or B = 180-arcsin(0.634283)
B = 39.366826 or B = 140.633174
B = 39.4 or B = 140.6
Each result is approximate.
Arcsine is the same as inverse sine. Denoted as the button on many calculators.
Please make sure that your calculator is set to degree mode.
Desmos for instance is set by default to radian mode. Click the wrench in the upper right corner to access the settings to change from radians to degrees.

If B = 39.4, then,
A+B+C = 180
A = 180-B-C
A = 180-39.4-80
A = 60.6 degrees
The result is between 0 and 180, which means we have a valid angle.
At least one triangle is possible at this point.

If B = 140.6, then,
A+B+C = 180
A = 180-B-C
A = 180-140.6-80
A = -40.6 degrees
This result is not between 0 and 180, so we toss this scenario.
B = 140.6 is not possible.
We have proven that only one triangle is possible.

Use the Law of Sines again.
sin(A)/a = sin(C)/c
sin(60.6)/a = sin(80)/5.9
5.9*sin(60.6) = a*sin(80)
a = 5.9*sin(60.6)/sin(80)
a = 5.219457
a = 5.2

--------------------------------------------------------

Summary
AnglesSides
A = 60.6 degrees (approximate)
B = 39.4 degrees (approximate)
C = 80 degrees		a = 5.2 (approximate)
b = 3.8
c = 5.9

Diagram

Only one triangle is possible.

More practice
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204424.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204365.html