# How to Use the Sine Rule

This guide has everything you need on how to use the sine rule. Before tackling this topic, you should be very confident using right-angled trigonometry (SOHCAHTOA). The sine rule works with any triangle but, most often, it is used when the triangle isn’t right-angled.

First, let’s remind ourselves how to label a triangle when working with the sine rule – this is the same as with the cosine rule. We label the vertices with capital letters and then label the opposite side with the corresponding lower-case letter. The label for the vertex is usually used for the angle as well. So, side $$a$$ is opposite angle $$A$$ and side $$b$$ is opposite angle $$B$$ and so on. Often, the vertices are labelled for you in exam questions, but you can add labels if you need to – just make sure they are really clear.

There are two versions of the sine rule that can be used:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

Usually, we use the left-hand equation when looking for missing sides and the right-hand equation when looking for missing angles. Since these equations are so similar, it is worth learning both.

Example 1
ABC is a triangle such that angle ABC = 89°, angle BCA = 23° and AB = 4.3cm. Find the length of AC, giving your answer to 3 significant figures.

First, you should always label the triangle. Then, you should decide which of the two formats is easier to use. In this case, we are looking for a missing side so we will use the one on the left.

Although the sine rule has 3 parts to it, we only need two parts. In this case, the parts that use $$b$$ and $$c$$ (as we have $$c$$ and are looking for $$b$$).

$\frac{b}{\sin B} = \frac{c}{\sin C}$

Now, we can substitute in the values we know and solve the equation to find $$b$$.

\begin{aligned} \frac{b}{\sin(89°)} &= \frac{4.3}{\sin(23°)} \\ b &= \frac{4.3\sin(89°)}{\sin(23°)} \\& = 11.0cm\end{aligned}

Example 2
ABC is a triangle such that angle ACB = 67°, BC = 9.7cm and AB = 11.3cm. Find angle BAC. Give your answer to 1 decimal place

As in example 1, you should start by labelling the triangle. In this case, we are looking for a missing angle so we will use the right-hand format.

$\frac{\sin A}{a} = \frac{\sin C}{c}$

Now, we can substitute in the values we know and solve the equation to find $$A$$.

\begin{aligned} \frac{\sin A}{9.7} &= \frac{\sin(67°)}{11.3} \\ \sin A &= \frac{9.7\sin(67°)}{11.3} \\& = 0.790…\end{aligned}

Remember, the inverse of $$\sin$$ is $$\sin^{-1}$$

\begin{aligned} A &= \sin^{-1}(0.790…) \\ &= 52.2°\end{aligned}

Now that you know how to use the sine rule, you’re probably ready for more revision – you can find more of our blogs here! You can also subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too.