**Note** – The Sine rule is considered a **higher** topic on most exam boards.

As well as the usual trigonometric formulas for right angled triangles, there’s also a handy little formula that you can use for * any* triangle. This is known as the

**Sine Rule**and can be used to find the size of the angles or the lengths of a side. The formula is as follows:

or alternatively

Where the uppercase letters correspond to the angle, and the lower case letter is the side opposite it. It doesn’t matter which one you use, but I recommend the first one if you’re working out an angle, and the second one if you’re working out the length of a side. This is just to simplify your calculations.

The Sine Rule can be used when you have either **two sides with an angle opposite to one of the sides** or **one side and any two angles**.

**Example 1**

Find the angle **y** in the above diagram,

Using the first of the formulas above, we get .

Rearranging gives .

We then use the inverse sine to get to 2dp.

**Example 2**

Find the length **x** in the above diagram.

This time we are going to be using the second of the above formulas. Subbing in the above values we get .

Rearranging gives to 2dp.

You have now seen how to find both sides and angles using the sine rule.

See also the Cosine Rule