Question

For the ambiguous case triangle that follows, solve both cases for all angles and sides. Include sketches of both triangles. Given: ΔABC, where ∠A = 23°, a = 11.9 cm, b = 16.8 cm.

Answer 1

To solve the ambiguous case triangle, use the Law of Sines. Given angle A = 23°, side a = 11.9 cm, and side b = 16.8 cm, two possible solutions can be found: Case 1 with angles A ≈ 23°, B ≈ 2.53°, C ≈ 154.47°, and sides a = 11.9 cm, b = 16.8 cm, c ≈ 21.1 cm; Case 2 with angles A ≈ 23°, B ≈ 131.47°, C ≈ 25.53°, and sides a = 11.9 cm, b = 16.8 cm, c ≈ 7.3 cm.

Step-by-step explanation:

To solve the ambiguous case triangle, we will use the Law of Sines. Since we have the side a = 11.9 cm, angle A = 23°, and side b = 16.8 cm, we can find angle C using the equation: sin(C) / c = sin(A) / a. Plugging in the known values, we get sin(C) / c = sin(23°) / 11.9. Solving for sin(C) and then finding angle C gives us two possible solutions:

1. Case 1: Angle C ≈ 154.47°, which makes angle B ≈ 2.53°.
2. Case 2: Angle C ≈ 25.53°, which makes angle B ≈ 131.47°.

Using the Law of Sines again, we can find side c for both cases:

1. Case 1: c ≈ 21.1 cm.
2. Case 2: c ≈ 7.3 cm.

Therefore, for Case 1, the triangle has angles A ≈ 23°, B ≈ 2.53°, C ≈ 154.47°, and sides a = 11.9 cm, b = 16.8 cm, c ≈ 21.1 cm. For Case 2, the triangle has angles A ≈ 23°, B ≈ 131.47°, C ≈ 25.53°, and sides a = 11.9 cm, b = 16.8 cm, c ≈ 7.3 cm.