The Sine Rule

Use the sine rule, pairing each side with the angle facing it, to find a missing side or a missing angle in a triangle with no right angle.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1 · 47 real past-paper questions below

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The Sine Rule

In every triangle, each side divided by the sine of the angle facing it gives the same number.

Formula

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

Side $a$ faces angle $A$ , side $b$ faces angle $B$ . You only ever use two of the three parts.

To find an angle, turn the rule upside down. It is the same rule, with the sines on top.

Formula

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

Use this form when the unknown is an angle. Use the first form when the unknown is a side.

Rule

Use it when you know a side and the angle facing it, a matching pair, plus one more side or angle.

Key idea

In about 1852, surveyors in India, including the mathematician Radhanath Sikdar, found that 'Peak XV' was the highest mountain on Earth — with no one climbing it. They measured angles from distant stations and solved triangles. We now call Peak XV Mount Everest.

Finding a Side

The 3-mark question: two angles and one side. Find the side you want, with the unknown side on top. Here $A = 4 0^{\circ}$ , $B = 7 5^{\circ}$ and $a = 8$ cm. Find $b$ .

Worked example

1
$a = 8 faces A = 4 0^{\circ}, b = ? faces B = 7 5^{\circ}$ — List what you know. The 8 and the 40° face each other, so they are the matching pair. We want $b$ , which faces 75°.
2
$\frac{b}{s i n 7 5 ^{\circ}} = \frac{8}{s i n 4 0 ^{\circ}}$ — Write the rule with only these two parts. The unknown side goes on top, so one multiplication will leave it alone.
3
$b = \frac{8 s i n 7 5 ^{\circ}}{s i n 4 0 ^{\circ}}$ — Multiply both sides by $sin 7 5^{\circ}$ to get $b$ alone.
4
$b = \frac{8 \times 0.9659}{0.6428} = 12.0 cm$ — Use the calculator in degree mode. Do the whole sum in one go and round at the end, to 3 significant figures.
5
$7 5^{\circ} > 4 0^{\circ} \Rightarrow b > a$ — Check: the bigger angle faces the longer side. 12.0 is longer than 8, so the answer makes sense.

Common mistake

Pairing a side with the wrong angle. Writing $8 sin 4 0^{\circ} \div sin 7 5^{\circ}$ gives 5.32. The side facing the bigger angle must be longer.

Key idea

Always write the line with the numbers in before you use the calculator. If you press a wrong key, that line still earns marks for working.

Finding an Angle

The 3-mark question: two sides and an angle facing one of them. Put the sines on top. Here $a = 9$ cm, $A = 6 2^{\circ}$ and $b = 7$ cm. Find $B$ .

Worked example

1
$\frac{s i n B}{7} = \frac{s i n 6 2 ^{\circ}}{9}$ — We want an angle, so put the sines on top. 7 faces $B$ and 9 faces 62°.
2
$sin B = \frac{7 s i n 6 2 ^{\circ}}{9}$ — Multiply both sides by 7 to get $sin B$ alone.
3
$sin B = 0.6867$ — Work out $7 \times sin 6 2^{\circ} \div 9$ on the calculator. Keep all the digits it shows for the next line.
4
$B = sin^{- 1} (0.6867) = 43. 4^{\circ}$ — Inverse sine undoes the sine and gives the angle.
5
$b < a \Rightarrow B < A$ — Check: 7 is shorter than 9, so $B$ must be smaller than 62°. 43.4° passes. A shorter side faces a smaller angle.
6
$C = 18 0^{\circ} - 6 2^{\circ} - 43. 4^{\circ} = 74. 6^{\circ}$ — If the question asks for the third angle, take the two known angles from 180°, because angles in a triangle add up to 180°.

Key idea

Sometimes an obtuse answer, 180° minus this one, also fits. That is The Ambiguous Case of the Sine Rule.

Find the Third Angle First

Often you are given two angles and a side that does not face either of them. You have no matching pair yet. Find the third angle first. Here $A = 5 2^{\circ}$ , $B = 7 1^{\circ}$ and $c = 12$ cm. Find $a$ .

Worked example

1
$C = 18 0^{\circ} - 5 2^{\circ} - 7 1^{\circ} = 5 7^{\circ}$ — Angles in a triangle add up to 180°. $C$ faces the 12 cm side, so now 12 and 57° are a matching pair.
2
$\frac{a}{s i n 5 2 ^{\circ}} = \frac{12}{s i n 5 7 ^{\circ}}$ — Write the rule with the unknown side $a$ on top, and the pair on the right.
3
$a = \frac{12 s i n 5 2 ^{\circ}}{s i n 5 7 ^{\circ}}$ — Multiply both sides by $sin 5 2^{\circ}$ to get $a$ alone.
4
$a = \frac{12 \times 0.7880}{0.8387} = 11.3 cm$ — Use the calculator in degree mode, to 3 significant figures.
5
$5 2^{\circ} < 5 7^{\circ} \Rightarrow a < c$ — Check: $a$ faces the smaller angle, so it should be shorter than 12 cm. 11.3 is.

Common mistake

Using the 71° angle with the 12 cm side. The 12 cm side faces $C$ , not $B$ , so it pairs with 57°.

Which Method? Read the Question

Find a matching pair first.

You see	You do
A pair, find a side	side on top
A pair, find an angle	sines on top
Two angles given	the third is 180° minus them
The angle answer could be obtuse	The Ambiguous Case of the Sine Rule
No pair at all	The Cosine Rule
Area of the shape	Area of a Triangle Using Sine
Upper or lower bound	Calculating with Bounds

Key idea

Check the answer: the longest side faces the largest angle.

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