Sine rule - ambiguous case visualiser
The sine rule ambiguous case happens when the information you are given can produce more than one possible triangle. This can occur when you know two sides and an angle that is not between them.
Use the interactive visualiser below to change the angle and side lengths. Watch how the circle intersects the ray from A and see when no triangle, one triangle or two different triangles are possible.
The visualiser also links the geometry to the sine rule calculation, helping to explain why a second possible angle can appear.
Interactive maths visualisation
Explore the sine rule ambiguous case
Change the known angle and side lengths. The dashed circle shows every possible position of B that keeps BC = a. Watch what happens where the circle meets the ray from A.
Current geometry
Two triangles are possible
The circle cuts the ray twice, so there are two valid positions for B.
What does the sine rule give?
B = sin⁻¹(0.819) = 55.0°
Second possible angle: 180° − 55.0° = 125.0°
Where is the ambiguous range?
Here, h = b sin A = 5.74 cm. Two triangles are possible when 5.74 < a < 10.00.
At a = h, the circle just touches the ray, so there is exactly one right-angled triangle.
Finished? Why not try our sine rule ambiguous case worksheet?