__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 ÞSine rule**

**Example:**

Calculate the length, in cm, of

*AB*.

*Solution:**ÐACB =*180

^{o}– (50

^{o}+ 70

^{o}) = 60

^{o}

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

**Example:**

Calculate Ð

*ACB.*

*Solution:***(C) Case of ambiguity (2 possible triangles)**

**Example**

Calculate Ð

*ACB*, θ

*.*

*Solution:*Two possible triangle with these measurement

*AB*

*=*26cm

*BC =*28 cm

*Ð*

*BAC*

*=*54

^{o}

$\begin{array}{l}\frac{26}{\mathrm{sin}\theta}=\frac{28}{\mathrm{sin}{54}^{o}}\\ \mathrm{sin}\theta =0.7512\\ \theta ={\mathrm{sin}}^{-1}0.7512\\ \theta ={48.7}^{o},{180}^{o}-{48.7}^{o}\\ \theta ={48.7}^{o}\text{(Acuteangle)},\text{}{131.3}^{o}\text{(Obtuseangle)}\end{array}$