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 1204641: 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°
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%7Ba+=+7
b+=+3
c=5.5
alpha=107°
beta=+24°
gamma=49°

highlight%282%29.+%0D%0A%0D%0A%7B%7B%7Ba+=+7
b+=+3
c=7.3
alpha=72.3°
beta=+24°
gamma=83.7°


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

Law of Sines
sin(alpha)/a = sin(beta)/b
sin(alpha)/7 = sin(24)/3
sin(alpha) = 7*sin(24)/3
sin(alpha) = 0.949052
alpha = arcsin(0.949052) or alpha = 180-arcsin(0.949052)
alpha = 71.631972 or alpha = 108.368028
alpha = 71.6 or alpha = 108.4

If alpha = 71.6, then
alpha + beta + gamma = 180
71.6 + 24 + gamma = 180
gamma = 180 - 71.6 - 24
gamma = 84.4
This result is between 0 and 180, so we have a valid angle.


If alpha = 108.4, then
alpha + beta + gamma = 180
108.4 + 24 + gamma = 180
gamma = 180 - 108.4 - 24
gamma = 47.6
This result is between 0 and 180, so we have a valid angle.

We have two possible triangles.
This is the SSA (side side angle) case. Specifically it is the ambiguous case because it's not clear which triangle to pick.

Let's find the missing side c when
  • alpha = 71.6 (approx)
  • beta = 24
  • gamma = 84.4 (approx)
We can use the Law of Sines to do so
sin(gamma)/c = sin(beta)/b
sin(84.4)/c = sin(24)/3
3*sin(84.4) = c*sin(24)
c = 3*sin(84.4)/sin(24)
c = 7.340578
c = 7.3

Now let's find the missing side c when
  • alpha = 108.4 (approx)
  • beta = 24
  • gamma = 47.6 (approx)
We can use the Law of Sines
sin(gamma)/c = sin(beta)/b
sin(47.6)/c = sin(24)/3
3*sin(47.6) = c*sin(24)
c = 3*sin(47.6)/sin(24)
c = 5.44668
c = 5.4

To summarize we have two possible triangles
Triangle 1Triangle 2
Anglesalpha = 71.6 (approx)
beta = 24
gamma = 84.4 (approx)		alpha = 108.4 (approx)
beta = 24
gamma = 47.6 (approx)
Sidesa = 7
b = 3
c = 7.3 (approx)		a = 7
b = 3
c = 5.4 (approx)



I used GeoGebra to confirm each answer.

More practice with solving triangles
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204423.html