You know sides a = 7, b = 10, and angle A = 30° in a triangle. You compute sin(B) = b·sin(A)/a ≈ 0.714. How many valid triangles exist?
AExactly one — the Law of Sines always gives a unique solution
BPossibly two — sin(B) < 1, so B could be either acute or obtuse
CNone — side b is longer than side a, so the triangle cannot be constructed
DInfinitely many — SSA configurations are always indeterminate
This is the ambiguous case (SSA). When sin(B) < 1 and angle A is acute, two values of B are possible: one acute (B ≈ 45.6°) and one obtuse (B ≈ 134.4°). Since both give A + B < 180°, both produce valid triangles. The key insight is that sin is not one-to-one on [0°, 180°]: sin(θ) = sin(180°−θ), so a computed sine value corresponds to two possible angles. Checking whether each yields a valid angle sum determines how many triangles exist.
Question 2 Multiple Choice
You are solving triangle ABC where angle A = 50°, angle B = 70°, and side c = 15. What is the correct first step using the Law of Sines?
AUse the Law of Cosines first: c² = a² + b² − 2ab·cos(C)
BFind C = 60° from the angle sum, then set up a/sin(50°) = 15/sin(60°)
CNo law applies — you need at least two sides to solve any triangle
DApply the Law of Sines but first verify the triangle inequality
This is an ASA configuration (two angles and a non-included side — but once you find C, it becomes effectively AAS). First, find the third angle: C = 180° − 50° − 70° = 60°. Now all three angles and one side are known, so the Law of Sines gives a unique solution: a = 15·sin(50°)/sin(60°). This clean case produces no ambiguity because two angles fully determine the triangle's shape.
Question 3 True / False
In the Law of Sines, the ratio a/sin(A) equals twice the radius of the triangle's circumscribed circle.
TTrue
FFalse
Answer: True
A beautiful consequence of the derivation: the common ratio a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius — the radius of the circle passing through all three vertices. This follows from the inscribed angle theorem. The Law of Sines is secretly a statement about the circumscribed circle, which is why the same formula appears in circle geometry.
Question 4 True / False
The ambiguous case (SSA) in the Law of Sines is a flaw in the formula — it means the Law of Sines gives an incorrect or incomplete answer for certain inputs.
TTrue
FFalse
Answer: False
The ambiguous case is not a formula flaw — it reflects a genuine geometric fact. When you specify two sides and an angle opposite one of them (SSA), there may be zero, one, or two geometrically valid triangles. A short side opposite an acute angle can swing to two different positions and still close the triangle. The Law of Sines correctly captures all possible solutions; the 'ambiguity' is in the problem setup, not the formula. Sketching the triangle before computing is the best way to see which sub-case applies.
Question 5 Short Answer
In an SSA configuration, when does the ambiguous case produce exactly zero valid triangles? Explain geometrically.
Think about your answer, then reveal below.
Model answer: Zero triangles exist when sin(B) > 1, which is mathematically impossible. Geometrically, this occurs when the side opposite the given angle is too short to reach across and close the triangle — the arc swept by the swinging side falls short of the base line entirely. For example, if angle A = 30° and a is very short relative to b, the side a cannot bridge the gap to complete the triangle. Checking sin(B) = b·sin(A)/a > 1 is the algebraic test for this geometric impossibility.
Grounding the algebraic condition (sin(B) > 1) in the geometric picture (the swinging side falls short) turns a seemingly arbitrary rule into an insight. Students who only memorize the algebraic test often cannot explain why it means 'no triangle exists.'