Sine Rule (AA SL)
Once a triangle loses its right angle, you need a rule that links every side to the angle opposite it - that's the sine rule. This page covers the formula in both its side-finding and angle-finding forms, when it applies instead of the cosine rule, and the mistakes that trip students up under exam pressure. It's part of the broader Triangles (Sine/Cosine Rules) topic.
15 questions on this sub-topic.
The sine rule
Covered under IB syllabus reference SL3.2: the sine rule (not including the ambiguous case). The ratio form is given in the formula booklet - you just need to know which side of it to work with.
Finding a side
\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
Use directly when you know one full angle-side pair and one more angle. Cross-multiply to isolate the unknown side.
Finding an angle
\(\sin B=\dfrac{b\sin A}{a}\)
The same rule, rearranged with the unknown angle's sine on top. IB SL exams never test the ambiguous case, so there's only ever one sensible answer to report.
Given SAS or SSS instead? See Cosine rule - or check the full topic page for GDC guidance.
Worked examples
In triangle \(ABC\), \(A = 40^\circ\), \(B = 75^\circ\) and side \(a = 8\) cm.
Find side \(b\).
Worked solution
Set up the sine rule with the known angle-side pair: \(\dfrac{b}{\sin 75^\circ}=\dfrac{8}{\sin 40^\circ}.\) M1
Rearrange: \(b=\dfrac{8\sin 75^\circ}{\sin 40^\circ}.\) A1
Evaluate: \(b\approx 12.0\) cm. A1
In triangle \(ABC\), \(a = 9\), \(b = 7\), \(B = 38^\circ\), where angle \(A\) is acute.
Find the value of \(A\).
Worked solution
Rearrange the sine rule for \(\sin A\): \(\sin A=\dfrac{a\sin B}{b}=\dfrac{9\sin 38^\circ}{7}=0.7916.\) M1
Since \(A\) is acute, \(A=\sin^{-1}(0.7916)\approx 52.3^\circ.\) A1
Show that the area of a triangle with sides \(a,b\) and included angle \(C\) equals \(\dfrac{a^2\sin B\sin C}{2\sin A}.\)
Worked solution
\(b = \dfrac{a\sin B}{\sin A}.\) M1 A1
Area \(= \tfrac12 ab\sin C\) M1 \(= \tfrac12 a\cdot\dfrac{a\sin B}{\sin A}\cdot\sin C = \dfrac{a^2\sin B\sin C}{2\sin A}.\) A1 AG ∎
Common mistakes
- Reaching for the sine rule with SAS data. If you only know two sides and the angle between them, there's no angle-side pair to plug into the sine rule - you need the cosine rule instead.
- Pairing the wrong side with the wrong angle. Each fraction must match a side to the angle directly opposite it - swap a pairing and the whole equation is wrong, even though the numbers look plausible.
- Rounding an intermediate angle before the final step. Truncating \(\sin^{-1}(0.7916)\) to two significant figures partway through a multi-part question can shift the final answer outside the accepted tolerance.
Ready to practise properly?
15 sine-rule questions, marked instantly like the real exam.
Quick answers
When should I use the sine rule instead of the cosine rule?
Use the sine rule when you know a matching angle-side pair (an angle and the side directly opposite it) plus one more piece of information - either another angle, or another side opposite a known angle.
What is the sine rule formula?
\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\), where each side is paired with the angle directly opposite it. IB SL examinations do not test the ambiguous case.