Sine Rule - Ambiguous Case Worksheet - GeoGebra - Free Printable

Let me work through each of these 6 triangle problems using the Law of Sines. The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C).

Problem 1: Find all possible values for ABC
- Given: AC = 8.1, BC = 4.5, BAC = 29.7°
- Using Law of Sines: BC/sin(A) = AC/sin(B)
- 4.5/sin(29.7°) = 8.1/sin(B)
- sin(B) = 8.1 × sin(29.7°) / 4.5
- sin(B) = 8.1 × 0.4955 / 4.5
- sin(B) = 4.0136 / 4.5
- sin(B) = 0.8919
- B = arcsin(0.8919) = 63.1° or 180° - 63.1° = 116.9°
- Check if both are valid: A + B must be < 180°
- Case 1: 29.7° + 63.1° = 92.8° < 180° ✓
- Case 2: 29.7° + 116.9° = 146.6° < 180° ✓
- Both solutions are valid!

Problem 2: Find all possible values for ∠ABC
- Given: AC = 6.4, BC = 4.1, ∠BAC = 38.7°
- Using Law of Sines: BC/sin(A) = AC/sin(B)
- 4.1/sin(38.7°) = 6.4/sin(B)
- sin(B) = 6.4 × sin(38.7°) / 4.1
- sin(B) = 6.4 × 0.6252 / 4.1
- sin(B) = 4.0013 / 4.1
- sin(B) = 0.9759
- B = arcsin(0.9759) = 77.4° or 180° - 77.4° = 102.6°
- Check:
- Case 1: 38.7° + 77.4° = 116.1° < 180° ✓
- Case 2: 38.7° + 102.6° = 141.3° < 180° ✓
- Both valid!

Problem 3: Find all possible values for ∠ABC
- Given: AC = 6.7, BC = 4.5, ∠BAC = 40.6°
- Using Law of Sines: BC/sin(A) = AC/sin(B)
- 4.5/sin(40.6°) = 6.7/sin(B)
- sin(B) = 6.7 × sin(40.6°) / 4.5
- sin(B) = 6.7 × 0.6508 / 4.5
- sin(B) = 4.3604 / 4.5
- sin(B) = 0.9690
- B = arcsin(0.9690) = 75.7° or 180° - 75.7° = 104.3°
- Check:
- Case 1: 40.6° + 75.7° = 116.3° < 180° ✓
- Case 2: 40.6° + 104.3° = 144.9° < 180° ✓
- Both valid!

Problem 4: Find all possible values for ∠BCA
- Given: AB = 5, BC = 4, ∠BAC = 38.7°
- Using Law of Sines: BC/sin(A) = AB/sin(C)
- 4/sin(38.7°) = 5/sin(C)
- sin(C) = 5 × sin(38.7°) / 4
- sin(C) = 5 × 0.6252 / 4
- sin(C) = 3.126 / 4
- sin(C) = 0.7815
- C = arcsin(0.7815) = 51.4° or 180° - 51.4° = 128.6°
- Check:
- Case 1: 38.7° + 51.4° = 90.1° < 180° ✓
- Case 2: 38.7° + 128.6° = 167.3° < 180° ✓
- Both valid!

Problem 5: Find all possible values for ∠BCA
- Given: AC = 5.7, BC = 7.2, ∠ABC = 33.7°
- Using Law of Sines: AC/sin(B) = BC/sin(A)
- Wait, I need to find ∠BCA (angle C).
- AC/sin(B) = AB/sin(C)... hmm, let me reconsider.
- Actually: AC/sin(B) = BC/sin(A) gives us angle A.
- Then angle C = 180° - A - B
- 5.7/sin(33.7°) = 7.2/sin(A)
- sin(A) = 7.2 × sin(33.7°) / 5.7
- sin(A) = 7.2 × 0.5548 / 5.7
- sin(A) = 4.0 / 5.7
- sin(A) = 0.6998
- A = arcsin(0.6998) = 44.4° or 180° - 44.4° = 135.6°
- Check:
- Case 1: A = 44.4°, then C = 180° - 44.4° - 33.7° = 101.9° ✓
- Case 2: A = 135.6°, then C = 180° - 135.6° - 33.7° = 10.7° ✓
- Both valid!

Wait, let me re-examine Problem 5. We're finding ∠BCA directly.
- Using Law of Sines: AB/sin(C) = AC/sin(B) = BC/sin(A)
- We know AC = 5.7, BC = 7.2, B = 33.7°
- To find ∠C, we need side AB, which we don't have.
- So we find ∠A first: AC/sin(B) = BC/sin(A)
- 5.7/sin(33.7°) = 7.2/sin(A)
- sin(A) = 7.2 × sin(33.7°) / 5.7 = 0.6998
- A = 44.4° or 135.6°
- Then C = 180° - A - B
- C = 101.9° or 10.7°

Problem 6: Find all possible values for ∠BAC
- Given: AC = 2.2, BC = 5.8, ∠ABC = 15°
- Using Law of Sines: AC/sin(B) = BC/sin(A)
- 2.2/sin(15°) = 5.8/sin(A)
- sin(A) = 5.8 × sin(15°) / 2.2
- sin(A) = 5.8 × 0.2588 / 2.2
- sin(A) = 1.501 / 2.2
- sin(A) = 0.6823
- A = arcsin(0.6823) = 43.0° or 180° - 43.0° = 137.0°
- Check:
- Case 1: 43.0° + 15° = 58.0° < 180° ✓
- Case 2: 137.0° + 15° = 152.0° < 180° ✓
- Both valid!

Let me verify my calculations more carefully using Python.