Law of Sines

The law of sines says: in every triangle the ratio of a side to the sine of its opposite angle is constant, a/sin α = b/sin β = c/sin γ. You pick two of the three quotients and solve for the sought quantity. The prerequisite is that at least one pair of a side and its opposite angle is completely known. Example: c = 10, γ = 80°, α = 40° gives a = c·sin α/sin γ = 10·0.643/0.985 ≈ 6.53. Typical cases: ASA/AAS (two angles and one side; the third angle comes from the 180° angle sum) and SSA (two sides and a non-included angle, beware ambiguity). If no complete pair exists, as in SAS or SSS, the law of cosines is the right tool.

If two sides and a NON-included angle are given, there can be two different triangles. The reason: the law of sines only yields sin β, and because sin(180° − β) = sin β, two angles fit this value, one acute and one obtuse. Example: a = 7, b = 9, α = 45° gives sin β = 9·sin 45°/7 ≈ 0.909, so β₁ ≈ 65.4° or β₂ ≈ 114.6°. Both are valid if the angle sum leaves the third angle positive: γ₁ ≈ 69.6° and γ₂ ≈ 20.4°, so here two triangles really exist. Checking scheme: compute β₂ = 180° − β₁ and test whether α + β₂ < 180°. The situation is always unique when the side opposite the given angle is the longer one.

Set up the appropriate pair of quotients and solve for the side: b = a·sin β/sin α. For this you need a complete pair (a and α) plus the opposite angle β of the sought side. Example: a = 8, α = 45°, β = 60° gives b = 8·sin 60°/sin 45° = 8·0.866/0.707 ≈ 9.80. If the opposite angle of the sought side is missing, compute it first via the angle sum: γ = 180° − α − β = 75°, then c = 8·sin 75°/sin 45° ≈ 10.93. Check with the ordering rule: the larger side always lies opposite the larger angle; here the chain a < b < c grows matching 45° < 60° < 75°. Set the calculator to DEG and do not round intermediate values too early.

The common value of the three quotients has a geometric meaning: a/sin α = b/sin β = c/sin γ = 2R, where R is the radius of the circumscribed circle, the circle through all three vertices. This follows from the inscribed angle theorem: the angle α appears over the chord a, and the chord length is a = 2R·sin α. This extended form answers bonus questions elegantly: from a = 6 and α = 30° it follows immediately that 2R = 6/0.5 = 12, the circumcircle has radius 6. Conversely it explains why the law of sines holds at all: all three sides are chords of the same circle, so their lengths scale uniformly with the sines of their inscribed angles.

Because the inverse function arcsin only outputs values between 0° and 90° (for positive inputs). A triangle angle can be obtuse, however, and because sin(180° − β) = sin β the calculator sees no difference between β and its supplement. Example: if the true angle is 114.6°, arcsin(0.909) still displays 65.4°. Therefore, after every arcsin in the law of sines you must actively check whether 180° minus the display could be meant instead: does the angle sum fit? Does the largest side lie opposite the largest angle? Does the sketch say obtuse or acute? As a fallback strategy you can compute large angles with the law of cosines, whose arccos covers the full range up to 180° uniquely, or always determine the smaller angles first.