Choosing between the sine rule and the cosine rule

These two rules let you find the missing sides and angles of any triangle, not just right-angled ones. The sine rule, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, links each side to the sine of the angle opposite it, so it works when you can pair a known side with its opposite angle.

The cosine rule, $a^{2} = b^{2} + c^{2} - 2bc\cos A$, links three sides and one angle, so it works when the sine rule cannot even get started. Between them they solve triangles from the standard data sets: two angles and a side, two sides and the included angle, or all three sides.

Choosing the right rule is the whole skill, the arithmetic afterwards is short. It also underpins area, bearings and 3-D problems later in the syllabus.

Look at what the triangle gives you. If you have two sides and the angle between them (SAS), or all three sides (SSS), start with the cosine rule, the sine rule has no complete side–angle pair to begin from.

If instead you have a side together with the angle opposite it, two angles and any side (AAS/ASA), or two sides and a non-included angle (SSA), use the sine rule.

A quick test settles it: can you name a side and the angle directly opposite it, both known? If yes, the sine rule is available; if no, reach for the cosine rule first, find one more part, and the sine rule usually opens up for the rest.

Keep the calculator in degree mode, and remember that with two sides and a non-included angle the sine rule can give two possible answers, the ambiguous case.

1. 1

   Label the triangle

   Mark sides $a, b, c$ opposite angles $A, B, C$; a side and its opposite angle share the letter.

2. 2

   List what you know

   Write down the given sides and angles and mark clearly what you must find.

3. 3

   Choose the rule

   SAS or SSS $\rightarrow$ cosine rule; a side with its opposite angle $\rightarrow$ sine rule.

4. 4

   Write the rule with your letters

   For an angle by the cosine rule use $\cos A = \frac{b^{2} + c^{2} - a^{2}}{2bc}$; for a side use $a^{2} = b^{2} + c^{2} - 2bc\cos A$.

5. 5

   Substitute and solve

   Put in the numbers, keep the calculator in degree mode, and solve for the unknown.

6. 6

   Check it is reasonable

   The largest angle faces the longest side, and the three angles add to $180^{\circ}$.

• Trying the sine rule on SAS or SSS data, where there is no complete side–angle pair to start from, use the cosine rule first.
• The ambiguous case (SSA): when the sine rule gives an angle, a second, obtuse angle $180^{\circ} - \theta$ may also fit, check whether it is possible.
• A sign slip in the cosine rule, it is $-\,2bc\cos A$; when $A$ is obtuse, $\cos A$ is negative and that term becomes an addition.
• Pairing a side with the wrong angle, side $a$ must go with angle $A$, the one opposite it.
• Leaving the calculator in radian mode, so $\cos 60^{\circ}$ or $\sin 60^{\circ}$ is evaluated wrongly.

The step that decides the whole question is the choice of rule, and the classic slip is forcing the sine rule onto two sides and their included angle, where it cannot even start. In one-to-one lessons our teachers give you one quick test,'can I name a side and the angle opposite it, both known?', and you write the answer down before choosing: yes means sine rule, no means cosine rule.

We drill the cosine-rule sign, $-\,2bc\cos A$, and flag the ambiguous SSA case so a second obtuse angle is never quietly missed. Because SPM Add Math is marked analytically, quoting the correct rule and substituting properly already earns method marks even if a rounding step slips.

Lessons are taught in English, while the SPM paper is set bilingually. Every teacher on spmaddmath.com.my is experienced, with lessons online at your own pace.