# Sine Rule

Whenever we are discussing a triangle and its properties in general, the notation we’ll assume will correspond to the followed triangle:

A triangle follows these basic properties:

**(a)** \(A + B + C = \pi \)

**(b)** Triangle inequality: \(a + b > c,\;b + c > a,\;c + a > b\)

These are many other simple properties of a triangle that most of you might be familiar with. In the following pages, we’ll discuss these and other properties, presenting proofs wherever necessary, and discussing applications of these properties.

**Property - 1: Sine Rule**

The sides of a triangle are proportional to the sines of the angles opposite to them:

\[\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\]

We discuss the justification for the case that \(\Delta ABC\) is acute:

The sine rule follows from extending this. Similarly, we can prove the sine rule for an obtuse angled or right angled triangle.

Note that the area of the triangle \(\Delta \) is

\[\Delta = \frac{1}{2} \times BC \times AD = \frac{1}{2}\;\;ab\;{\text{sin}}\;C = \frac{1}{2}ac\sin B\]

This leads to the expression for area of a triangle:

\[\boxed{\;\Delta = \frac{1}{2}ab\sin C = \frac{1}{2}\;bc\;\sin A = \frac{1}{2}ca\sin B\;}\]

This is a widely used relation.