Question

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 larger than ∠B2.) a = 31, c = 44, ∠A = 31°

Answer 1

Explanation:

the law of sine says

a/sinA = b/sinB = c/sinC

with a, b, c being the sides, and A, B, C being the corresponding opposite angles.

so, in our case we know this ratio via a and A :

31/sin(31) = 60.18972482...

c/sinC = 44/sinC = 31/sin(31)

31×sinC = 44×sin(31)

sinC = sin(31)×44/31 = 0.731021784...

C = 46.97212232...° ≈ 47°

or

C ≈ 180 - 47 ≈ 133°

both angles are possible in combination with A = 31°, because the sum of all angles in a triangle is always 180°, and even 133 + 31 = 164° allows a third angle (16°).

so, with C1 = 47°, we get B1 = 180 - 47 - 31 = 102°.

with C2 = 133°, we get B2 = 180 - 133 - 31 = 16°.

b1/sinB1 = b1/sin(102) = 31/sin(31)

b1 = 31×sin(102)/sin(31) = 58.87443492... ≈ 58.9

b2/sinB2 = b2/sin(16) = 31/sin(31)

b2 = 31×sin(16)/sin(31) = 16.5905366... ≈ 16.6