The sine rule is used to find unknown lengths or angles in non right-angled triangles. It is particularly useful when one side and two angles are given, or two sides and one angle are given.

## Labelling convention The capital letters denote the interior angle at that point. In other words, B is the angle ABC, and A is the angle BAC. The lower-case letters are the lengths of the side opposite to the corresponding angle. That means that a, in this case is opposite to A and represents the length BC.

## The sine rule

The sine rule states that for a triangle ABC, $\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)} = \dfrac {c}{sin(C)}$

### Application

Find x in the following triangle: Using the sine rule, $\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)}$ $\dfrac{x}{sin(50)} = \dfrac{8}{sin(30)}$ $x = \dfrac{8sin(50)}{sin(30)}$ $x = 12.26$

### The ambiguous case

In some occasions, using the sine rule directly will give the wrong angle output.

Example:

Find angle A in the following triangle: Using the sine rule, $\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)}$ $\dfrac{30}{sin(A)} = \dfrac{20}{sin(40)}$ $sin(A) = \dfrac{30}{20}sin(40)$ $A = sin^{-1} 0.96$
Using a calculator, $A = 74.62^o$

However, it is clear from the image that A is an obtuse angle. The inverse sine function on the calculator only gives the acute angle. If the angle is obtuse, then subtraction from $180^o$ is necessary in order to obtain the right answer.

In this case, $A = 180 - 74.62 = 105.38^o$

This ambiguous situation can be illustrated as below where A* = the angle given by the calculator.