Step by Step
1
The formula
a/sin(A) = b/sin(B) = c/sin(C) — each side divided by the sine of its opposite angle gives the same ratio, for any triangle (not just right triangles).
2
When to use it: AAS or ASA
Use the Law of Sines when you know two angles and any side (whether the side is between the angles or not) — since knowing two angles lets you find the third by subtracting from 180°, giving you a full angle-opposite-side pair to work with.
3
When to use it: SSA — the ambiguous case
Also use the Law of Sines when you know two sides and a non-included angle (SSA). This case is called "ambiguous" because, depending on the specific values, there could be zero, one, or two valid triangles satisfying the given information.
4
Why SSA is ambiguous
When solving SSA, you may get two possible angles from the inverse sine function (since sine is positive in both Quadrant I and Quadrant II) — you have to check whether both, one, or neither of these possible angles actually produces a valid triangle (angles summing to less than 180°).
Applied Walkthrough
1
A triangle has angle A = 40°, angle B = 60°, and side a = 10. Find side b.
2
Since two angles and one side are known (AAS), the Law of Sines applies directly: a/sin(A) = b/sin(B).
3
Set up the ratio: 10/sin(40°) = b/sin(60°).
4
Solve for b: b = 10·sin(60°)/sin(40°) ≈ 10·(0.866)/(0.643) ≈ 13.47.
Exam Application
Exams test whether you can recognize when the Law of Sines applies (AAS, ASA, or SSA), and specifically whether you can recognize and correctly resolve the ambiguous SSA case by checking for zero, one, or two valid triangles.
⚠ Common Trap
The most common trap is treating the SSA case like any other — forgetting to check whether the calculated angle actually produces a valid triangle, or missing that a second valid triangle might also exist using the supplementary angle.
✓ Quick Self-Check
1. What is the Law of Sines formula?
a/sin(A) = b/sin(B) = c/sin(C).
Tap to reveal / hide
2. What triangle information types call for the Law of Sines?
AAS, ASA, or SSA (the ambiguous case).
Tap to reveal / hide
3. Why is the SSA case called "ambiguous"?
Because it can produce zero, one, or two valid triangles, depending on the specific values given.
Tap to reveal / hide
4. Given angle A = 40°, angle B = 60°, and side a = 10, find side b.
b ≈ 13.47.
Tap to reveal / hide
5. Why can the SSA case yield two possible angles?
Because sine is positive in both Quadrant I and Quadrant II, so the inverse sine function can correspond to two different angles.
Tap to reveal / hide