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 = 78, b = 109, ∠A = 24°
Find :
∠B1 = °
∠B2 = °
∠C1 = °
∠C2 = °
c1 =
c2 =
sinB/109 = sin24°/78
That gives you B
Then C = 180-(A+B)
then c/sinC = a/sinA
To solve for all possible triangles that satisfy the given conditions using the Law of Sines, follow these steps:
1. Determine the missing side length using the Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we are given a = 78 and ∠A = 24°. We need to find b and c.
b/sin(B) = a/sin(A)
b/sin(B) = 78/sin(24°)
b = sin(B) × (78/sin(24°))
c/sin(C) = a/sin(A)
c/sin(C) = 78/sin(24°)
c = sin(C) × (78/sin(24°))
2. Solve for ∠B1:
∠B1 = arcsin(b1/a) = arcsin(b1/78)
3. Solve for ∠B2:
∠B2 = 180° - ∠B1
4. Solve for ∠C1:
∠C1 = arcsin(c1/a) = arcsin(c1/78)
5. Solve for ∠C2:
∠C2 = 180° - ∠C1
6. Calculate c1:
Plug the values of ∠C1 and ∠B1 into the Law of Sines:
c1/sin(∠C1) = a/sin(∠A)
c1 = sin(∠C1) × (a/sin(∠A))
7. Calculate c2:
Plug the values of ∠C2 and ∠B2 into the Law of Sines:
c2/sin(∠C2) = a/sin(∠A)
c2 = sin(∠C2) × (a/sin(∠A))
Now, let's plug in the given values and calculate the answers:
a = 78, b = 109, ∠A = 24°
Step 1:
b = sin(B) × (78/sin(24°))
c = sin(C) × (78/sin(24°))
Step 2:
∠B1 = arcsin(b1/78)
Calculate the value of b1 for ∠B1 using the given values.
Step 3:
∠B2 = 180° - ∠B1
Step 4:
∠C1 = arcsin(c1/78)
Calculate the value of c1 for ∠C1 using the given values.
Step 5:
∠C2 = 180° - ∠C1
Step 6:
c1 = sin(∠C1) × (78/sin(24°))
Step 7:
c2 = sin(∠C2) × (78/sin(24°))
Calculate the numerical values using a calculator and round them to one decimal place.