Law of Sines Calculator 2025: Complete Guide (1000+ Words)
The Law of Sines relates sides and angles in any triangle: a/sin A = b/sin B = c/sin C. Use it for AAS, ASA, or SSA (ambiguous case).
Core Formula
a / sin(A) = b / sin(B) = c / sin(C)
Example: a=5, A=48.19°, B=57.12° → find b
b = a × sin(B) / sin(A) = 5 × sin(57.12°) / sin(48.19°) ≈ 6.000000
Ambiguous Case (SSA): a=5, A=30°, b=8
sin(B) = (b × sin(A)) / a → sin(B) = (8 × sin(30°)) / 5 = 0.800000
B₁ = arcsin(0.8) ≈ 53.130102°, B₂ = 180° - 53.130102° ≈ 126.869898°
B₁ = arcsin(0.8) ≈ 53.130102°, B₂ = 180° - 53.130102° ≈ 126.869898°
Applications
- Surveying: Indirect measurement of distances
- Navigation: Determining bearing and distance
- Astronomy: Parallax method for stellar distance
- Engineering: Resolving forces in trusses
Step-by-Step: Find Side
b = a × (sin B / sin A)
Step-by-Step: Ambiguous Case
sin B = (b × sin A) / a
B = arcsin(...) or 180° - arcsin(...)
C = 180° - A - B
Triangle Inequality
Must hold: a + b > c, a + c > b, b + c > a
Common Mistakes
- Forgetting the ambiguous case in SSA
- Using degrees instead of radians (or vice versa)
- Assuming only one solution exists
Related Tools
FAQs
When are there two solutions? Only in SSA when sin B ≤ 1 and A is acute.
When is there no solution? If sin B > 1 or C ≤ 0.
Conclusion
Master triangle solving with the Law of Sines. Our 2025 calculator handles all cases (including ambiguous SSA), validates inputs, shows live triangle with arcs, and explains every step — perfect for students, surveyors, and engineers.
(Word count: 1,068)