10.1 The Sine Rule

10.1 The Sine Rule

In a triangle ABC in which the sides BC, CA and AB are denoted by a, b, and c as shown, and A, B, C are used to denote the angles at the vertices A, B, C respectively,

The sine rule can be used when
(i) two sides and one non-included angle or
(ii) two angles and one opposite side are given.

(A) If you know 2 angles and 1 side Þ Sine rule


Calculate the length, in cm, of AB.
ÐACB = 180o – (50o + 70o) = 60o
AB sin 60 o = 4 sin 50 o AB= 4×sin 60 o sin 50 o AB=4.522 cm

(B) If you know 2 sides and 1 angle (but not between them) Þ Sine rule


Calculate ÐACB.
28 sin 54 o = 26 sinACB sinACB= 26×sin 54 o 28 sinACB=0.7512 ACB= 48.7 o

(C) Case of ambiguity (2 possible triangles)


Calculate ÐACBθ.

Two possible triangle with these measurement
AB = 26cm         BC = 28 cm           Ð BAC = 54o
26 sinθ = 28 sin 54 o sinθ=0.7512 θ= sin 1 0.7512 θ= 48.7 o , 180 o 48.7 o θ= 48.7 o  (Acute angle),  131.3 o  (Obtuse angle)