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.