Two sides and a non-included angle can determine zero, one or two triangles. If the sine rule gives sin B=k, check the range first. For 0<k<1, test both B_1=sin-1k and B_2=180 -B_1. Each candidate must leave 180 -A-B>0. Example Let A=30, a=5 cm, b=8 cm. Then sin B= 8sin30 5=0.8. The candidates are B 53.1 or 126.9, giving C 96.9 or 23.1. Both are valid, so there are two triangles. If the calculation gives sin B>1, no triangle exists. If sin B=1, then B=90; the supplementary candidate is the same angle, not a second answer. Always use unrounded candidate angles for subsequent side calculations. An obtuse angle already given elsewhere can rule out an obtuse candidate, because two obtuse angles cannot fit inside a triangle. Report the information for each valid triangle separately.

Two sides and a non-included angle can determine zero, one or two triangles. If the sine rule gives \(\sin B=k\), check the range first. For \(0<k<1\), test both \(B_1=\sin^{-1}k\) and \(B_2=180^\circ-B_1\). Each candidate must leave \(180^\circ-A-B>0\).

Let \(A=30^\circ\), \(a=5\text{ cm}\), \(b=8\text{ cm}\). Then \(\sin B=\dfrac{8\sin30^\circ}{5}=0.8\). The candidates are \(B\approx53.1^\circ\) or \(126.9^\circ\), giving \(C\approx96.9^\circ\) or \(23.1^\circ\). Both are valid, so there are two triangles.

If the calculation gives \(\sin B>1\), no triangle exists. If \(\sin B=1\), then \(B=90^\circ\); the supplementary candidate is the same angle, not a second answer. Always use unrounded candidate angles for subsequent side calculations.

An obtuse angle already given elsewhere can rule out an obtuse candidate, because two obtuse angles cannot fit inside a triangle. Report the information for each valid triangle separately.

Checking two possible triangles

Checking two possible triangles