## Tuesday, 7 February 2017

### The Sine Rule, the Cosine Rule and the Area of a Triangle

To understand the sine and cosine rules, we need to view a triangle like this:

In these triangles, the small case letters are the sides and the capital equivalents to those letters represent the angles directly opposite them.

### The Sine Rule

If, in a question, you are asked to find an angle in a triangle, and you are given two sides and another angle, you need to use the sine rule for angles.

Let's imagine that you are given a question like this and you have to find the angle:
Use the first diagram and label the sides and their corresponding angles. Here the missing angle would be A and 7cm would be a, as that is the side opposite the missing angle. 81° would be B, meaning that 18cm would be b.

Putting these values in the formula shows us that:

If you are asked to find a missing side in a triangle with two angles and one side, you have to use the sine rule for sides, as illustrated below.

The method used here is similar to the method illustrated above.

### The Cosine Rule

If you are given a question where you have to find the missing side given three sides, you would have to use the cosine rule for angles.

Let's imagine you are given a question like this where you had to find the angle:

If the missing angle is A, then 9cm would be a. 10cm would then be b and 11cm would then be c.

Putting these values in the formula shows us that:

If you are asked to find a missing side with two sides and an angle, you would have to use the cosine rule of sides.

### The Area of a Triangle

If you have an angle and two sides in a triangle and you are asked to find the area, you have to use this formula:

Let us consider the second example on this page. A is 50.5°, b is 10cm and c is 11cm.

Putting these values in the formula gives us: