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 1204652: 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. Below, enter your answers so that 𝛼1 is smaller than 𝛼2.)
a = 7,

b = 3,

𝛽 = 24°
a1=
gamma1=
c1=
a2=
gamma2=
c2=
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a+=+7
b+=+3
beta=+24°
using Law of Cosine, we have
b=sqrt%28a%5E2%2Bc%5E2-2%2Aa%2Ac%2Acos%28beta%29%29
3=sqrt%287%5E2%2Bc%5E2-2%2A7%2Ac%2Acos%2824%29%29
9=49%2Bc%5E2-14%2Ac%2Acos%2824%29
9=49%2Bc%5E2-14%2Ac%2A0.9135454576426
9=49%2Bc%5E2-12.7896c
0=49%2Bc%5E2-12.7896c-9
c%5E2-12.7896c%2B40=0
using quadratic formula and got:
c=5.5
or
c=7.3


then angle alpha is
if c=5.5

alpha=cos%5E-1%28%28b%5E2%2Bc%5E2-a%5E2%29%2F%282bc%29%29
alpha=cos%5E-1%28%283%5E2%2B5.5%5E2-7%5E2%29%2F%282%2A3%2A5.5%29%29
alpha=107
or
if c=7.3
alpha=cos%5E-1%28%283%5E2%2B7.3%5E2-7%5E2%29%2F%282%2A3%2A7.3%29%29
alpha=72.3


then angle gamma is
gamma=180-%2824%2B107%29
gamma=49°
or
gamma=180-%2824%2B72.3%29
gamma=83.7°

so, there are two triangles:
highlight%281%29.+%0D%0A%0D%0A%7B%7B%7Ba1+=+7
b1+=+3
c1=5.5
alpha%281%29=107°
beta%281%29=+24°
gamma%281%29=49°

highlight%282%29.+%0D%0A%0D%0A%7B%7B%7Ba2+=+7
b2+=+3
c2=7.3
alpha%282%29=72.3°
beta%282%29=+24°
gamma%282%29=83.7°

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

This was answered previously at this link:
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204641.html
Please let me know if you have questions about any of the steps.