Sine Rule (AI HL)
The sine rule connects a triangle's angles to the sides opposite them, letting you solve triangles that have no right angle at all. It's the natural choice whenever you're handed an angle-side opposite pair plus one more piece of information. This page covers the formula, two worked examples, and the mistakes worth avoiding. It's part of the broader Geometry & Trigonometry topic.
11 questions on this sub-topic.
The formula
Covered under IB syllabus reference SL3.2, which pairs the sine rule with the cosine rule and the \(\tfrac12ab\sin C\) area formula (this section excludes the ambiguous case, so only one triangle is possible per question). All three sit in the formula booklet.
Sine rule
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\]
Needs an angle and its opposite side, plus one more known side or angle - solve for whichever is missing from the matching pair.
✓ In the formula bookletCosine rule
\[c^2=a^2+b^2-2ab\cos C\]
Steps in when there's no opposite pair available - two sides with the included angle, or all three sides.
✓ In the formula bookletNeed the full syllabus wording, formula-booklet table and GDC steps? See Geometry & Trigonometry.
Worked examples
Two lighthouses \(A\) and \(B\) are 15 km apart. A ship \(S\) is 9 km from \(A\). The angle \(ASB = 110^{\circ}\).
Find the angle \(SAB\), to 3 significant figures.
Worked solution
\(\dfrac{\sin(SBA)}{AS} = \dfrac{\sin(ASB)}{AB} \Rightarrow \sin(SBA) = \dfrac{9\sin 110^\circ}{15} = 0.5638.\) M1
\(\angle SBA \approx 34.3^\circ.\) A1
\(\angle SAB = 180^\circ - 110^\circ - 34.3^\circ\) M1
\(\approx 35.7^\circ.\) A1
In triangle \(ABD\), \(AB = 12\), \(\angle ABD = 50^{\circ}\), \(\angle ADB = 60^{\circ}\). Find \(AD\) (3 significant figures).
Worked solution
\(\angle BAD = 180 - 50 - 60\) M1
\(\angle BAD = 70^{\circ}.\) A1
\(\dfrac{AD}{\sin 50^{\circ}} = \dfrac{12}{\sin 60^{\circ}} \Rightarrow AD = \dfrac{12\sin 50^{\circ}}{\sin 60^{\circ}}\) M1
\(AD \approx 10.6.\) A1
Common mistakes
- Forcing the sine rule onto a two-sides-included-angle question. Without an angle-side opposite pair, the sine rule has nothing to start from - that's a sign to switch to the cosine rule instead.
- Rounding the first angle before the angle-sum step. If a question needs a second angle afterwards (as in a triangle-splitting problem), round only the final answer - carrying a rounded intermediate value through an angle sum shifts the last significant figure.
- Mismatching sides and angles in the ratio. \(a\) must be the side opposite \(A\), \(b\) opposite \(B\), and so on - relabelling a triangle without keeping this pairing straight sends the whole substitution wrong.
- Missing the ambiguous case for a non-included angle. Given two sides and a non-included angle, \(\sin^{-1}\) can return two valid triangles - check whether the obtuse supplementary angle also fits the given information before ruling it out.
Ready to practise properly?
11 sine-rule questions, marked instantly like the real exam.
Quick answers
When do I use the sine rule?
Use the sine rule when you know an angle and its opposite side, plus one more known side or angle. It works for any triangle, not just right-angled ones.
What is the sine rule formula?
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\), where each side sits opposite its matching angle. It's given in the formula booklet.