Question

Solve the triangle using the Law of Sines. (Assume b=3,∠A=30∘ , and ∠C=120∘. Round the lengths to two decimal places.)

a =
c =
∠B =

Answer 1

To solve the triangle using the Law of Sines with given parameters b=3, ∠A=30°, and ∠C=120°, we find ∠B to be 30°, side a to be 3, and side c, when rounded, to be approximately 5.20.

Step-by-step explanation:

To solve the triangle using the Law of Sines, we'll first need to find the remaining angle, ∠B, since a triangle's angles sum up to 180 degrees. Given that ∠A is 30° and ∠C is 120°, we can calculate ∠B as 180° - ∠A - ∠C, which equals 180° - 30° - 120° = 30°.

Using the Law of Sines, ∠A / a = ∠B / b = ∠C / c, and given that b = 3, we can find the sides a and c by using the proportion:

• sin(30°) / a = sin(30°) / 3
• sin(120°) / c = sin(30°) / 3

Solving for a gives us a = 3 * sin(30°) / sin(30°) = 3.

To find c, we calculate c = 3 * sin(120°) / sin(30°). Using a calculator, we find c to be approximately 3 * 0.866 / 0.5 = 5.196, which when rounded is c = 5.20.

So the sides are a = 3, b = 3, and c = 5.20. The angles are ∠A = 30°, ∠B = 30°, and ∠C = 120°.