Sine Rule (AI SL)
The sine rule links each side of a triangle to the sine of the angle opposite it, and it's the tool of choice whenever a question gives you a matching side-angle pair. This page covers using it to find a missing side and a missing angle. It's part of the broader Triangle Trigonometry topic.
15 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL3.2: the sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). This section of the syllabus does not include the ambiguous case, so any question with two possible triangles will tell you which one to find.
Sine rule for a side
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}\]
Rearrange to make the unknown side the subject once you know its opposite angle.
✓ In the formula bookletSine rule for an angle
\[\dfrac{\sin A}{a}=\dfrac{\sin B}{b}\]
The same rule flipped upside down - useful when the unknown is an angle rather than a side.
✓ In the formula bookletNeed the wider syllabus context and GDC settings for trig? See Triangle Trigonometry's GDC guidance.
Worked examples
In triangle \(ABC\), \(A=42^\circ\), \(B=78^\circ\) and \(a=8\) cm.
Find the length of side \(b\), to 3 significant figures.
Worked solution
Sine rule.
\(\dfrac{b}{\sin B}=\dfrac{a}{\sin A}\Rightarrow b\) M1
\(=\dfrac{8\sin78^\circ}{\sin42^\circ}\approx\dfrac{8\times0.9781}{0.6691}\approx11.7\) cm. A1
In triangle \(ABC\), \(a=11\) cm, \(b=9\) cm and \(A=62^\circ\).
Find angle \(B\).
Worked solution
\(\dfrac{\sin B}{9}=\dfrac{\sin62^\circ}{11}\Rightarrow\sin B=\dfrac{9\sin62^\circ}{11}\) M1
\(\approx0.7236.\) A1
\(B=\sin^{-1}(0.7236)\approx46.3^\circ.\) A1
R1
Common mistakes
- Reaching for the sine rule when the cosine rule is needed. If you know two sides and the angle between them (or all three sides), the sine rule can't be set up - you need the cosine rule instead.
- Pairing a side with the wrong angle. Each side in the sine rule must sit over the sine of the angle directly opposite it - \(a\) with \(A\), \(b\) with \(B\), and so on.
- Assuming an angle found from \(\sin^{-1}\) is automatically correct. Even though the IB AI SL syllabus excludes the full ambiguous case, always check the angle found is consistent with the rest of the triangle (e.g. angles summing to \(180^\circ\), the largest angle facing the longest side).
- Rounding an intermediate angle before using it in the next step. Carry full calculator accuracy through the sine rule and only round the final answer - rounding the angle early can shift the last significant figure of the side that follows.
Ready to practise properly?
15 sine-rule questions, marked instantly like the real exam.
Quick answers
When can I use the sine rule?
Use the sine rule whenever you have a complete side-angle pair (a side and the angle directly opposite it) plus one more side or angle. If you only have two sides and the angle between them, use the cosine rule instead.
Does the IB AI SL syllabus include the ambiguous case of the sine rule?
No - SL3.2 explicitly excludes the ambiguous case. Where a question does have two possible triangles, it will tell you which one to use (for example, by asking for the acute angle).