The Ambiguous Case of the Sine Rule

Spot when two sides and an angle not between them allow two different triangles, find both angles with the sine rule, and pick the right one when the question says acute or obtuse.

In the exam for Edexcel Further Pure Maths 4PM1, Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 5 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Two Triangles From the Same Facts

Sine has the same value for an angle and for 180° minus that angle: $sin 5 2^{\circ} = sin 12 8^{\circ}$ . So when the sine rule finds an angle, there may be two answers.

This can happen only when you know two sides and an angle that is not between them.

Rule

If the known angle is acute and faces the shorter of the two sides, check both answers: $x$ and $18 0^{\circ} - x$ .

Finding Both Angles

$A = 3 5^{\circ}$ , $a = 8$ cm and $b = 11$ cm. Find angle $B$ . The 35° faces the shorter side, so expect two answers.

Worked example

1
$\frac{s i n B}{11} = \frac{s i n 3 5 ^{\circ}}{8}$ — The sine rule, with the angle you want on top. Side 11 faces B and side 8 faces the 35°.
2
$sin B = \frac{11 \times s i n 3 5 ^{\circ}}{8}$ — Multiply both sides by 11 to get sin B on its own.
3
$sin B = 0.7887$ — Work out the right-hand side on the calculator.
4
$B = 52. 1^{\circ}$ — Press inverse sine. The calculator only ever gives the acute angle.
5
$B = 18 0^{\circ} - 52. 1^{\circ} = 127. 9^{\circ}$ — An angle and 180° minus it have the same sine, so 127.9° is the second possible answer.
6
$3 5^{\circ} + 127. 9^{\circ} = 162. 9^{\circ}$ — Test the second answer by adding the angle you know. A triangle has 180° in total, so this must stay under 180°.
7
$162. 9^{\circ} < 18 0^{\circ}$ — That leaves 17.1° for the third angle, so both triangles exist.

Common mistake

Stopping at the calculator answer. It only ever gives the acute angle.

When the Question Picks One

The 6-mark question tells you which one to use, such as "angle Q is obtuse". Then carry on to the area or a side.

Triangle $P QR$ : $P Q = 2$ cm, $P R = 5$ cm, $R = 2 0^{\circ}$ and $Q$ is obtuse. Find the area.

Worked example

1
$\frac{s i n Q}{5} = \frac{s i n 2 0 ^{\circ}}{2}$ — The sine rule. Side PR (5) faces Q, and side PQ (2) faces the 20° at R.
2
$sin Q = \frac{5 \times s i n 2 0 ^{\circ}}{2} = 0.8551$ — Multiply both sides by 5, then work it out.
3
$sin^{- 1} (0.8551) = 58. 8^{\circ}$ — The calculator gives the acute angle. This is the answer for the other triangle, not this one.
4
$Q = 18 0^{\circ} - 58. 8^{\circ} = 121. 2^{\circ}$ — The question says Q is obtuse, so use 180° minus the calculator angle.
5
$P = 18 0^{\circ} - 2 0^{\circ} - 121. 2^{\circ} = 38. 8^{\circ}$ — The angles in a triangle add to 180°. P is the one we do not know yet.
6
$Area = \frac{1}{2} (2) (5) sin 38. 8^{\circ}$ — The area formula needs two sides and the angle between them. P is between PQ and PR.
7
$Area = 3.13 cm^{2}$ — Work it out and give 3 significant figures.

Common mistake

Using the acute 58.8°. That is the other triangle, and gives 4.90 cm².

Same Area, Two Angles

Some questions say that a second triangle has the same area as the first. The two areas are equal, so you can find its angle. It can have two answers.

Triangle $A B C$ has $A B = 12$ cm, $A C = 8$ cm and $A = 3 5^{\circ}$ . Triangle $P QR$ has $P Q = 10$ cm, $QR = 7$ cm and the same area. Find the two possible values of angle $P QR$ .

Worked example

1
$\frac{1}{2} \times 12 \times 8 \times sin 3 5^{\circ} = 27.53$ — First find the area you have to match, using the two sides and the angle between them.
2
$\frac{1}{2} \times 10 \times 7 \times sin Q = 27.53$ — Q is the angle between PQ and QR, so write the same formula for PQR and make it equal to the first area.
3
$35 sin Q = 27.53$ — Half of 10 × 7 is 35.
4
$sin Q = \frac{27.53}{35} = 0.7866$ — Divide both sides by 35.
5
$Q = 51. 9^{\circ}$ — Inverse sine gives the acute angle first.
6
$Q = 18 0^{\circ} - 51. 9^{\circ} = 128. 1^{\circ}$ — The sine of an angle and of 180° minus it are the same, so 128.1° gives the same area. Both answers work, because there is no other angle to add.

Key idea

Keep the second answer even if the question does not mention a triangle with an obtuse angle. 51.9° and 128.1° both give the same area.

Which Method? Read the Question

Only one situation gives two triangles.

You see	You do
Two sides, angle not between, facing the shorter side	check both angles
"is obtuse" or "is acute"	pick that one
Two shapes with the same area	set the two areas equal, then find both angles
The known angle is obtuse	only one triangle
The known angle faces the longer side	only one triangle
Two sides and the angle between	The Cosine Rule
Any other sine rule question	The Sine Rule

Key idea

Test the second angle: add it to the angle you know. If the total reaches 180°, throw it away.

Practise the ambiguous case of the sine rule now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on the ambiguous case of the sine rule

Each with a full worked solution.

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.