Solving triangles with the sine and cosine rules

Most of the difficulty in this chapter is not the calculation, a scientific calculator does that in seconds, it is choosing the right rule before you start. Make that choice quickly and correctly and the rest is short.

Choose wrongly and you can burn ten minutes going nowhere.

There is a clean way to decide. Look at what the triangle gives you, and match it to the rule that fits:

• You have a side and the angle opposite it (a matching pair), plus one more piece, use the sine rule.
• You have two sides and the angle between them (SAS), use the cosine rule to find the third side.
• You have all three sides (SSS) and want an angle, use the cosine rule rearranged for the angle.

The single word that unlocks it is pair. If you already have a side opposite a known angle, the sine rule is almost always the faster tool.

If you do not have such a pair, the cosine rule is what gets you started.

The sine rule links each side of a triangle to the sine of the angle opposite it. Labelling the sides $a, b, c$ opposite angles $A, B, C$:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

You only ever use two of the three fractions at a time, the pair you can fill in. Suppose $A = 40^{\circ}$, the side opposite it is $a = 8$, and another side is $b = 10$.

To find angle $B$, pair the two fractions you can complete:

$$\frac{8}{\sin 40^{\circ}} = \frac{10}{\sin B} \;\Rightarrow\; \sin B = \frac{10 \sin 40^{\circ}}{8} \approx 0.8035$$

So $B \approx 53.5^{\circ}$. Notice how the rule turns a triangle problem into a single equation you can rearrange.

When you are finding a side rather than an angle, flip the fractions the other way up so the unknown sits on top, it saves a step.

The cosine rule is what you reach for when the sine rule has nothing to pair. In its first form it finds a side from two sides and the angle between them:

$$a^{2} = b^{2} + c^{2} - 2bc\cos A$$

For instance, with $b = 7$, $c = 5$ and the included angle $A = 60^{\circ}$: $a^{2} = 49 + 25 - 2(7)(5)\cos 60^{\circ} = 74 - 35 = 39$, so $a = \sqrt{39} \approx 6.24$. Rearranged, the same rule finds an angle when you know all three sides:

$$\cos A = \frac{b^{2} + c^{2} - a^{2}}{2bc}$$

With sides $a = 6$, $b = 7$, $c = 8$, the angle opposite the longest side is $\cos C = \dfrac{6^{2} + 7^{2} - 8^{2}}{2(6)(7)} = \dfrac{21}{84} = 0.25$, giving $C \approx 75.5^{\circ}$. One formula, two jobs, recognising which version you need is the whole skill.

Here is the trap that catches careful students. When you use the sine rule to find an angle, your calculator gives you an acute answer, but an obtuse angle can have the same sine.

Because $\sin\theta = \sin(180^{\circ} - \theta)$, there may be two valid triangles, and the question sometimes wants both.

In the earlier example, $\sin B \approx 0.8035$ gives $B \approx 53.5^{\circ}$, but also $B \approx 180^{\circ} - 53.5^{\circ} = 126.5^{\circ}$. Both can be genuine.

You keep an answer only if the three angles still add to $180^{\circ}$ with room to spare.

The chapter also gives you a neat way to find area without the perpendicular height, using two sides and the angle between them:

$$\text{Area} = \tfrac{1}{2}\,ab\sin C$$

For the triangle with $a = 6$, $b = 7$ and included angle $C \approx 75.5^{\circ}$, the area is $\tfrac{1}{2}(6)(7)\sin 75.5^{\circ} \approx 20.3$ square units. As always, the angle used must be the one between the two sides.

• Leaving the calculator in radian mode, for SPM degree work it must be in degrees, or every value will be wrong.
• Using the sine rule when there is no matching side-angle pair, instead of the cosine rule.
• Forgetting the obtuse possibility in the ambiguous case, or keeping an obtuse angle that makes the triangle impossible.
• Rounding part-way through: keep full accuracy on your calculator and round only the final answer.

This is a chapter where a student can know both rules perfectly and still lose marks, because the difficulty lives in the choice, the ambiguous case, and small calculator habits. Those are exactly the things a teacher can spot in real time.

Working one-to-one, a teacher watches which rule you reach for and why, catches the moment the obtuse answer is forgotten, and checks that your calculator is set up the way the paper expects. Our teachers are experienced; lessons are online and taught in English, while the SPM papers are set bilingually in Bahasa Melayu and English.

If solving triangles is where you slip, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We will not promise a grade, but we can help make the choice of rule feel automatic, which is where most of the marks in this chapter live.