Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B1 is smaller than ∠B2.)

a = 34, c = 43, ∠A = 39°

∠B1 =


°
∠B2 =


°
∠C1 =


°
∠C2 =


°
b1 =


b2 =

sinC/c = sinA/a

sinC/43 = sin39°/34 = 0.01851
sinC = .7959
so, C = 52.74° or 127.26°
Since A+B+C=180, B=88.26° or 13.74°

Now use those two values for B to get B, using

b/sinB = 1/0.01851 = 54.0265

Thank you so much! I am just trying to figure out how to solve for b1. I am confused as to how you got the 1 and why you used 0.01851 when you said to use the two values for B to get B (the two values are 88.26 and 13.74)

I mean, that for each value of c, there is a corresponding value of b.

Since B can be either 88.26 or 13.74,

b1/sin(88.26) = 54.0265, so b1=54.10
or
b2/sin(13.74) = 54.0265, so b2=12.86

To use the Law of Sines to solve for all possible triangles, we can follow these steps:

Step 1: Write down the given information:

a = 34 (opposite angle A),
c = 43 (opposite angle C),
∠A = 39°

Step 2: Find angle B using the law of sines:

sin(B) / a = sin(A) / c

sin(B) / 34 = sin(39°) / 43

sin(B) = (34 * sin(39°)) / 43

sin(B) ≈ 0.427

Taking the inverse sine of both sides:

B ≈ arcsin(0.427)

B ≈ 25.439° (rounded to one decimal place)

Step 3: Find the remaining angles:

∠C = 180° - ∠A - ∠B

∠C ≈ 180° - 39° - 25.439°

∠C ≈ 115.561° (rounded to one decimal place)

Step 4: Find the remaining sides:

We can use the Law of Sines again:

b / sin(B) = a / sin(A)

b / sin(25.439°) = 34 / sin(39°)

b ≈ (34 * sin(25.439°)) / sin(39°)

b ≈ 22.912 (rounded to one decimal place)

Now we have one possible triangle:

∠B1 = 25.439°
∠B2 = DNE (since there is only one possible value for ∠B)
∠C1 = 115.561°
∠C2 = DNE (since there is only one possible value for ∠C)
b1 = 22.912
b2 = DNE (since there is only one possible value for b)