Sine Rule

What is the Sine Rule?

The sine rule (or law of sines) is an equation which relates any triangle's side lengths to the sines of its angles. It is convention to label a triangle's sides with lower case letters, and its angles with the capitalised letter of the opposite side, as shown here. The sine rule is stated in terms of this labelling.

There are two ways of expressing the sine rule, both these lines are equivalent: $\frac{\sin(\text{A})}{\text{a}}=\frac{\sin(\text{B})}{\text{b}}=\frac{\sin(\text{C})}{\text{c}} \\[10pt] \frac{\text{a}}{\sin(\text{A})}=\frac{\text{b}}{\sin(\text{B})}=\frac{\text{c}}{\sin(\text{C})}$ When used in practice, you will only ever focus on two side-angle pairs, so will usually use a part of the sine rule such as $$\frac{\sin(\text{A})}{\text{a}}=\frac{\sin(\text{B})}{\text{b}}$$

When to use the Sine Rule

Use the sine rule when you're interested in two angle-side pairs, such as $$\text{a}$$, $$\text{A}$$, $$\text{b}$$ and $$\text{B}$$. You will know three pieces of information (lengths or angles) and be wanting to find the fourth. The sine rule can be used with any sort of triangle; there is no need for it to be right angled as with SOH CAH TOA.

Example 1: finding an angle

Find angle $$\text{A}$$, according to the diagram shown [not to scale].

Information we have: $$\text{a}=15$$, $$\text{c}=12$$ and $$\text{C}=50^o$$.
Sine rule portion to use: $\frac{\sin(\text{A})}{\text{a}}=\frac{\sin(\text{C})}{\text{c}}$ Substitute and rearrange: \begin{align} \frac{\sin(\text{A})}{15}&=\frac{\sin(50)}{12} \\[8pt] \sin(\text{A})&=\frac{15\sin(\text{50})}{\text{12}} \\[8pt] \sin(\text{A})&=0.9575556 \\[8pt] \text{A}&=\sin^{-1}(0.9575556) \\[8pt] \text{A}&=73.25^o \hspace{10pt}\text{[2dp]} \end{align}

Example 2: finding a side length

This question doesn't use $$\text{a}$$, $$\text{b}$$ or $$\text{c}$$. You can use any letters; with the key being to match angles to their opposite sides, regardless of your labels.

Find the length of side $$\text{x}$$, according to the diagram shown [not to scale].

Substitute and rearrange: \begin{align} \frac{\text{x}}{\sin(87)}&=\frac{27}{\sin(48)} \\[8pt] \text{x}&=\frac{27\sin(87)}{\sin(48)} \\[8pt] \text{x}&=36.28 \hspace{10pt}\text{[2dp]} \end{align}