3 · IB Maths AA
Ambiguous case of the sine rule
Fix angle A and side b, then change side a: watch side a swing from C and see whether it meets the base zero, one or two times, with both possible angles B calculated.
a / sin A = b / sin B
sin B = b sin A / a
B₂ = 180° − B₁
h = b sin A
C = 180° − A − B
c = a sin C / sin A
sin B = b sin A / a
B₂ = 180° − B₁
h = b sin A
C = 180° − A − B
c = a sin C / sin A
Try this
- Set A = 35°, b = 8.0 cm and slowly lower a from 9 cm to 4 cm. Note the exact values of a where the number of triangles changes and compare them with b and h.
- With two triangles showing, check that B₁ + B₂ = 180° and that triangle 2 is obtuse at B.
- Make A obtuse (e.g. 120°). Can you ever get two triangles? Explain using the sum of the angles.
Common error: the calculator's sin⁻¹ only returns the acute angle B₁, so always test B₂ = 180° − B₁ and keep it only if A + B₂ is less than 180°.