The Sine Rule
Master the sine rule for NSW Year 12 Mathematics Standard 2. In any triangle each side pairs with the angle opposite it, so \(\tfrac{a}{\sin A}=\tfrac{b}{\sin B}=\tfrac{c}{\sin C}\) lets you find a missing side or a missing angle in a triangle that is not right-angled.
You will learn to label a triangle correctly, choose the right pair of ratios, find an unknown side, find an unknown angle (including when it is obtuse), and give answers to the nearest degree or in degrees and minutes — a core Standard 2 skill for surveying, navigation and triangulation problems.
Theory
The sine rule relates each side of any triangle to the sine of the angle opposite it. This Year 12 Standard 2 (NSW) guide shows how to pair sides with opposite angles, find an unknown side or unknown angle in a non-right triangle, handle an obtuse angle, and give answers in degrees or degrees and minutes.
The sine rule works in any triangle, not just right-angled ones. In a triangle labelled \(ABC\), each side is named with the lower-case letter of the angle opposite it: side \(a\) faces angle \(A\), side \(b\) faces \(B\) and side \(c\) faces \(C\).
The rule says these side-over-sine ratios are all equal: \(\tfrac{a}{\sin A}=\tfrac{b}{\sin B}=\tfrac{c}{\sin C}\). You only ever use two of the three ratios at a time — the pair that contains your two known values and the one unknown.
Use it to find an unknown side (when you know a side and two angles) or an unknown angle (when you know two sides and a non-included angle). This is a core Year 12 Standard 2 (NSW) skill for surveying, navigation and other non-right-triangle problems. The ambiguous case is not examined in Standard 2.
For any triangle \(ABC\), the sine rule links each side to its opposite angle:
To find an angle, turn every ratio upside down so the sines are on top:
How to use the sine rule
- Label the triangle so each side matches the angle opposite it (\(a\) opposite \(A\), and so on).
- Pick the two ratios that contain your two knowns and the one unknown; ignore the third ratio.
- Set them equal. For a side keep the sides on top; for an angle flip the ratios so \(\sin\) is on top.
- Solve. For a side, multiply across; for an angle, apply \(\sin^{-1}\) with the calculator in degree mode.
- Round as asked — a decimal place for a length, the nearest degree, or degrees and minutes (\(\times 60\)). If two angles are given, first find the third with \(A+B+C=180^\circ\).
Pair \(b\) with \(B\) and \(a\) with \(A\); keep the sides on top.
| \(\dfrac{b}{\sin B}\) | \(=\) | \(\dfrac{a}{\sin A}\) |
| \(b\) | \(=\) | \(\dfrac{11\,\sin 68^\circ}{\sin 42^\circ}\) |
| \(b\) | \(=\) | \(\dfrac{11\times 0.9272}{0.6691}\) |
| \(\therefore\ AC\) | \(\approx\) | \(15.24\text{ cm}\) |
\(B\) is unknown, so flip the ratios to put \(\sin\) on top.
| \(\dfrac{\sin B}{b}\) | \(=\) | \(\dfrac{\sin A}{a}\) |
| \(\sin B\) | \(=\) | \(\dfrac{13\,\sin 64^\circ}{18}=0.6491\) |
| \(B\) | \(=\) | \(\sin^{-1}(0.6491)=40.48\ldots^\circ\) |
| \(\therefore\ B\) | \(\approx\) | \(40^\circ\) |
Find \(B\), then convert the decimal degree to minutes.
| \(\sin B\) | \(=\) | \(\dfrac{21\,\sin 53^\circ}{25}=0.6709\) |
| \(B\) | \(=\) | \(\sin^{-1}(0.6709)=42.13\ldots^\circ\) |
| \(0.13\ldots^\circ\) | \(=\) | \(0.13\ldots\times 60'\approx 8'\) |
| \(\therefore\ B\) | \(\approx\) | \(42^\circ 08'\) |
Get the acute value first, then take \(180^\circ-\sin^{-1}(\ldots)\).
| \(\sin B\) | \(=\) | \(\dfrac{17\,\sin 26^\circ}{9}=0.8280\) |
| \(\text{acute}\) | \(=\) | \(\sin^{-1}(0.8280)=55.90\ldots^\circ\) |
| \(B\) | \(=\) | \(180^\circ-55.90\ldots^\circ\) |
| \(\therefore\ B\) | \(\approx\) | \(124^\circ\) |
Common pitfalls
Frequently asked questions
What is the sine rule?
The sine rule says that in any triangle the ratio of each side to the sine of its opposite angle is the same: a over sin A equals b over sin B equals c over sin C. It lets you find a missing side or angle in a triangle that is not right-angled.
When do I use the sine rule instead of the cosine rule?
Use the sine rule when you have a matching side-and-opposite-angle pair: either a side and two angles (to find another side), or two sides and a non-included angle (to find another angle). If you only know three sides, or two sides and the angle between them, you need the cosine rule instead.
How do I rearrange the sine rule to find an angle?
Turn every ratio upside down so the sines are on top: sin A over a equals sin B over b. Substitute your known side and angle, work out the value of sin B, then apply the inverse sine on your calculator, which must be in degree mode.
How do I find a side with the sine rule?
Write two ratios with the sides on top, for example b over sin B equals a over sin A. Substitute the values you know and multiply across to make b the subject: b equals a times sin B divided by sin A. Then round as the question asks.
What if the triangle only gives me two angles?
First find the third angle using the fact that the angles of a triangle add to 180 degrees. Once you have an angle opposite a known side, you can pair it with any other side or angle and apply the sine rule.
When is the answer an obtuse angle?
The inverse sine on a calculator only returns the acute angle. The angle is obtuse when the side opposite it is the longest, or when the question tells you so; in that case take 180 degrees minus the acute value. Standard 2 excludes the ambiguous case, so you are always told which one to use.