# 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

Example:

Calculate the length, in cm, of AB.
Solution:
ÐACB = 180o – (50o + 70o) = 60o

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

Example:

Calculate ÐACB.
Solution:
$\begin{array}{l}\frac{28}{\mathrm{sin}{54}^{o}}=\frac{26}{\mathrm{sin}\angle ACB}\\ \mathrm{sin}\angle ACB=\frac{26×\mathrm{sin}{54}^{o}}{28}\\ \mathrm{sin}\angle ACB=0.7512\\ \angle ACB={48.7}^{o}\end{array}$

(C) Case of ambiguity (2 possible triangles)

Example

Calculate ÐACBθ.

Solution:
Two possible triangle with these measurement
AB = 26cm         BC = 28 cm           Ð BAC = 54o