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 ∠A1 is smaller than ∠A2.)
b = 74, c = 84, ∠B = 58°
∠A1 =
°
∠A2 =
°
∠C1 =
°
∠C2 =
°
a1 =
a2 =
sinC/c = sinB/b = sin58/74 = .01146
sinC = 84*0.1146 = .9626
So, C=74.29 or 105.71
Since A+B+C=180, A = 47.71 or 16.29
Now use
a/sinA = b/sinB = 1/.01146 = 87.259
a1/sin47.71 = 87.259
or
a2/sin16.29 = 87.259
Don't forget your Algebra I now that you're taking trig...
To use the Law of Sines to solve for the possible triangles that satisfy the given conditions, we can use the formula:
sin(A1) / a1 = sin(B) / b = sin(C1) / c
Let's solve for ∠A1 first.
1. Substitute the given values:
sin(A1) / a1 = sin(58°) / 74
2. Cross-multiply and solve for sin(A1):
sin(A1) = (a1 * sin(58°)) / 74
3. Take the inverse sine (sin^-1) of both sides to find the value of ∠A1:
∠A1 = sin^-1((a1 * sin(58°)) / 74)
Now, let's solve for ∠A2.
1. In a triangle, the sum of all interior angles is 180°. So, we can find ∠A2 using the formula:
∠A2 = 180° - ∠A1 - ∠B
2. Substitute the known values and calculate:
∠A2 = 180° - ∠A1 - 58°
Next, let's solve for ∠C1.
1. We can find ∠C1 using the formula:
∠C1 = 180° - ∠A1 - ∠B
2. Substitute the known values and calculate:
∠C1 = 180° - ∠A1 - 58°
Finally, let's solve for ∠C2.
1. In a triangle, the sum of all interior angles is 180°. So, we can find ∠C2 using the formula:
∠C2 = 180° - ∠A2 - ∠B
2. Substitute the known values and calculate:
∠C2 = 180° - ∠A2 - 58°
Lastly, let's find the values of a1 and a2 using the Law of Sines.
1. Substitute the known values into the Law of Sines equation:
sin(A1) / a1 = sin(B) / b
2. Cross-multiply and solve for a1:
a1 = (sin(A1) * b) / sin(B)
3. Similarly, solve for a2 using the same equation:
a2 = (sin(A2) * b) / sin(B)
Now, plug in the values into the equations and simplify to find ∠A1, ∠A2, ∠C1, ∠C2, a1, and a2. Then round your answers to one decimal place.