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:
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.
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 \(b\).
\(\begin{aligned} \frac{b}{\sin(89°)} &= \frac{4.3}{\sin(23°)} \\ b &= \frac{4.3\sin(89°)}{\sin(23°)} \\& = 11.0cm\end{aligned}\)
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 \(\sin^{-1}\)
\(\begin{aligned} A &= \sin^{-1}(0.790…) \\ &= 52.2°\end{aligned}\)
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