How to Use the Sine Rule
The sine rule lets you find a missing side or angle in any triangle, not just right-angled ones. This page shows you how to use the sine rule step by step, with worked examples, the ambiguous case where two triangles are possible, and unlimited auto-marked IGCSE practice across four rooms.
What is the sine rule?
SOHCAHTOA only works in right-angled triangles. The sine rule lets you find missing sides and angles in any triangle, whether or not it has a right angle.
Label each side with the lower-case letter of the angle opposite it: side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\). The sine rule links each side to the sine of its opposite angle.
The two forms
Use one form to find a side, and the flipped form to find an angle.
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
To find an ANGLE (you know two sides and a non-included angle):\[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\]
When to use the sine rule
The method
- Label the triangle: each side gets the letter of its opposite angle.
- Pick the two fractions that contain what you know and what you want (you only ever use two of the three at once).
- Choose the right form: side on top to find a side, sine on top to find an angle.
- Rearrange and solve, keeping full accuracy until the end, then round to 3 significant figures.
Worked examples
💡 Example 1: finding a side
In triangle ABC, angle A = 40°, angle B = 75°, and side b = 9 cm. Find side a.
Side on top (we want a side):
\[\frac{a}{\sin A} = \frac{b}{\sin B}\]
\[\frac{a}{\sin 40^\circ} = \frac{9}{\sin 75^\circ}\]
\[a = \frac{9}{\sin 75^\circ} \times \sin 40^\circ\]
\(a = 5.99\) cm (3 s.f.)
What's happening?
We know side \(b\) with its opposite angle \(B\), and we want side \(a\) whose opposite angle \(A\) is given. That is a matching pair, so the sine rule works.
Multiply both sides by \(\sin 40^\circ\) to get \(a\) on its own.
💡 Example 2: finding an angle
In triangle ABC, side b = 8 cm, side c = 6 cm, and angle C = 35°. Find angle B.
Sine on top (we want an angle):
\[\frac{\sin B}{b} = \frac{\sin C}{c}\]
\[\frac{\sin B}{8} = \frac{\sin 35^\circ}{6}\]
\[\sin B = \frac{\sin 35^\circ}{6} \times 8\]
\(B = \sin^{-1}(0.7648\ldots)\)
\(B = 49.9^\circ\) (3 s.f.)
What's happening?
We know side \(c\) with its opposite angle \(C\), and side \(b\), and we want angle \(B\). Another matching pair.
After finding \(\sin B\), use the inverse sine on your calculator to get the angle.
💡 Example 3: find the third angle first
In triangle PQR, angle P = 55°, angle Q = 65°, and side p = 12 cm. Find side r.
Angles in a triangle sum to \(180^\circ\):
\(R = 180^\circ - 55^\circ - 65^\circ\)
\(R = 60^\circ\)
\[\frac{r}{\sin R} = \frac{p}{\sin P}\]
\[r = \frac{12}{\sin 55^\circ} \times \sin 60^\circ\]
\(r = 12.7\) cm (3 s.f.)
What's happening?
We wanted side \(r\), but its opposite angle \(R\) was not given. Because two angles were known, the third comes from the angle sum, giving us the matching pair we need.
Extension: the ambiguous case
When you are given two sides and an angle that is not between them, there can sometimes be two different triangles that fit the same measurements. This is called the ambiguous case.
If a question says the angle is obtuse, or the diagram makes only one triangle possible, you take the matching answer. Otherwise, be ready to give both: if the sine rule gives \(B = 50^\circ\), the other possibility is \(180^\circ - 50^\circ = 130^\circ\).
🔑 Key points
- Each side is named after the angle opposite it.
- Side on top to find a side; sine on top to find an angle.
- You only use two of the three fractions at a time.
- You always need a complete side-and-opposite-angle pair.
- Keep full accuracy and round only at the end (3 s.f.).
⚠ Common pitfalls
- Using the sine rule when you only have two sides and the angle between them (use the cosine rule instead).
- Pairing a side with the wrong angle (it must be the opposite angle).
- Forgetting the inverse sine when finding an angle.
- Rounding partway through, which throws off the final answer.
- Calculator not in degree mode.