Solve a triangle from an opposite side–angle pair plus one more side or angle. The calculator keeps dependent fields synchronized, checks the 180° angle sum and explicitly reports whether SSA data produce zero, one or two triangles.
Complete the missing side or angle
Matching each side with its opposite angle
Each side must be paired with the angle directly opposite it. Once a/sin α is known, the same scale factor links b with β and c with γ.
Completing a triangle from a, α and β
With a = 10, α = 30° and β = 45°, b = 10·sin 45°/sin 30° ≈ 14.14 and γ = 105°. The calculator preserves the unrounded ratio when completing the remaining values.
Detecting the ambiguous SSA case
In SSA, arcsin returns an acute principal angle, but its supplement has the same sine. The second triangle exists only when that supplement and the known angle still sum to less than 180°.
When the sine law is the shortest route
Use it directly for ASA or AAS and for SSA when you are prepared to test the ambiguous case. If the known angle lies between two known sides (SAS), start with the cosine law instead.
Impossible data are not forced into a result. An angle outside (0°,180°), an angle sum of 180° or more, or an implied sine outside [−1,1] cannot describe a non-degenerate Euclidean triangle.
Questions about the sine rule
Why must one side be paired with its opposite angle?
Without a known pair, the common sine ratio cannot be established.
Can the same SSA data form two triangles?
Yes. An angle and its supplement have the same sine, provided both make a valid triangle.
Why must the angles total 180°?
That is the interior-angle sum of a Euclidean triangle; inputs violating it cannot describe one.
Why can SSA have two solutions but AAS cannot?
SSA fixes a height but may allow the unknown side to meet the base in two positions; two known angles already fix the third angle and remove that ambiguity.
