This is the last question on my pre-calc homework, I looked through the textbook and have no idea how to do this.
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 = 76 b = 108 ∠A = 21°
∠B1 = °
∠B2 = °
∠C1 = °
∠C2 = °
c1 =
c2 =
sinB/b = sinA/a
so,
sinB = 108*sin21°/76 = 0.50925
so, B = 30.6° or 149.4°
Since A+B+C = 180, that means
C is 128.4° or 9.6°
So, since
c/sinC = a/sinA
there are two choices for c, as well.
To solve this problem using the Law of Sines, you will need to use the following formula:
a/sin(A) = b/sin(B) = c/sin(C)
Given: a = 76, b = 108, ∠A = 21°.
To find ∠B1, we can use the Law of Sines formula:
b/sin(B1) = a/sin(A)
Plugging in the values:
108/sin(B1) = 76/sin(21°)
To solve for sin(B1), cross-multiply:
108 * sin(21°) = 76 * sin(B1)
Divide both sides by 108 to isolate sin(B1):
sin(B1) = (76 * sin(21°)) / 108
Using a calculator, find the value of sin(B1).
To find ∠B2, we can use the fact that the sum of the angles in a triangle is 180°. Since we know ∠A and ∠B1, we can find ∠B2:
∠B2 = 180° - ∠A - ∠B1
Plug in the known values:
∠B2 = 180° - 21° - ∠B1
Using the value of ∠B1 that you found earlier, you can now calculate ∠B2.
To find ∠C1, we can use the fact that the sum of the angles in a triangle is 180°:
∠C1 = 180° - ∠A - ∠B1
Plug in the known values:
∠C1 = 180° - 21° - ∠B1
Using the value of ∠B1 that you found earlier, you can now calculate ∠C1.
To find ∠C2, we can use the fact that the sum of the angles in a triangle is 180°:
∠C2 = 180° - ∠A - ∠B2
Plug in the known values:
∠C2 = 180° - 21° - ∠B2
Using the value of ∠B2 that you found earlier, you can now calculate ∠C2.
To find c1 and c2, we can use the Law of Sines formula:
c1/sin(C1) = a/sin(A)
c2/sin(C2) = a/sin(A)
Plugging in the values:
c1/sin(C1) = 76/sin(21°)
c2/sin(C2) = 76/sin(21°)
To solve for c1 and c2, cross-multiply and use a calculator to find their values.
Remember to round your answers to one decimal place.
I hope this explanation helps you solve the problem!