POP

G7.12 The Sine Rule

Image · 914 × 270

01 — Explanation

How do I label a non-right-angled triangle for the sine/cosine rules?

Use the opposite pairs convention: angle $A$ sits opposite side $a$ , angle $B$ opposite side $b$ , and so on. Every rule from here on is built around identifying these pairs correctly.

02 — Explanation

What's the sine rule, and which version do I use?

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \qquad\text{or equivalently}\qquad \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}

Use the side-on-top version when finding a missing side (less rearranging); use the sin-on-top version when finding a missing angle.

03 — Explanation

How do I know whether to use the sine rule at all?

You need a complete opposite pair — a known side and its directly opposite known angle. This happens in two situations: AAS (two angles + one side — the third angle follows from angles summing to $180°$ , instantly giving you a pair) and SSA (two sides + a non-included angle). If you don't have an opposite pair (SAS or SSS), you need the cosine rule instead, not the sine rule.

Worked example (AAS): In $\triangle PQR$ , $\angle Q=105°$ , $\angle R=43°$ , $PQ=7.2$ cm. Find $PR$ .

Side $r=PQ=7.2$ has its opposite angle $R=43°$ — a complete pair. Side $q=PR$ is opposite $Q=105°$ .

\frac{PR}{\sin105°}=\frac{7.2}{\sin43°} \Rightarrow PR=\frac{7.2}{\sin43°}\times\sin105°\approx10.2\text{ cm}

Exam Tip: Enter the whole expression into your calculator in one go — rounding $7.2\div\sin43°$ before multiplying introduces avoidable error.

04 — Explanation

What is the "ambiguous case", and when does it happen?

It only arises in SSA, when solving for an angle. $\sin^{-1}$ only ever returns the acute solution $\theta_1$ , but $\sin(180°-\theta_1)=\sin\theta_1$ too — so an obtuse angle $\theta_2=180°-\theta_1$ can be an equally valid second triangle.

If $\sin(\text{angle})>1$ when you calculate it: no triangle exists.
If the given side is at least as long as the adjacent side: only $\theta_1$ is valid.
Otherwise: both $\theta_1$ and $\theta_2$ may be valid — check whether $\theta_2$ plus the known angle still totals under $180°$ .

Exam Tip: Always check the question for the words "obtuse", "largest angle", or "two possible triangles" — that's the examiner telling you which case they want.

05 — Worked example

Worked exam-style question (ambiguous case)

In $\triangle LMN$ , $LM=10$ , $LN=8$ , $\angle LNM=30°$ ( $N$ opposite the side $LM=10$ ). The angle at $M$ is obtuse — find it.

Opposite pair: side $n=LM=10$ with $\angle N=30°$ . Unknown: $\angle M$ , opposite side $m=LN=8$ .
$\dfrac{\sin M}{8}=\dfrac{\sin30°}{10} \Rightarrow \sin M=0.4 \Rightarrow \theta_1=\sin^{-1}(0.4)\approx23.6°$ .
Since the question specifies obtuse: $\theta_2=180°-23.6°\approx156.4°$ .
Check: $N+\theta_2=30°+156.4°=186.4°$ ... if this exceeds $180°$ the obtuse case is impossible for that specific data — always verify before committing to an answer.

06 — Explanation

How do I decide between the sine rule, cosine rule, and area formula?

Given	Use	Finds
Right angle present	SOHCAHTOA / Pythagoras	side or angle
Two angles + one side (AAS)	Sine rule	missing side
Two sides + non-included angle (SSA)	Sine rule	missing angle (check ambiguous case)
Two sides + included angle (SAS)	Cosine rule	third side
All three sides (SSS)	Cosine rule	any angle
Two sides + included angle, asked for area	Area rule $\frac12ab\sin C$	area

07 — Common mistakes

Common mistakes to avoid

Using SOHCAHTOA on a non-right-angled triangle.
Finding an angle via the sine rule and stopping at the acute solution without checking whether the obtuse case applies.
Rounding an intermediate side or angle before using it in the next step.
Leaving the calculator in radian mode.