Topics

The sine rule

Covers AS91259

Labelling a triangle

The standard convention makes every formula readable:
- Capital letters ( $A$ , $B$ , $C$ ) label the angles — or the vertices.
- Lower-case letters ( $a$ , $b$ , $c$ ) label the sides.
- Each side is opposite the angle with the same letter. Side $a$ is opposite angle $A$ .

Label your own diagram this way before doing anything else. It converts the formulae from something to memorise into something to read off.

When the sine rule applies

Use the sine rule when you have a matching pair — a side and the angle opposite it.
The two situations:
- Two angles and any side (AAS or ASA)
- Two sides and an angle opposite one of them (SSA) — the ambiguous case

The formula

For finding a side:

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

For finding an angle, turn it upside down so the unknown is on top:

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

Both forms are the same equation. Choosing the one with the unknown in the numerator saves a rearranging step and a chance to slip.
You only ever use two of the three fractions at a time.

Finding a side

Identify the matching pair you know completely.
Write the two fractions, with the unknown side on top.
Multiply across and solve.
Worked through — in triangle $ABC$ , $A = 40°$ , $B = 75°$ and $a = 12$ cm. Find $b$ .
- Matching pair: $a = 12$ with $A = 40°$
- $\dfrac{b}{\sin 75°} = \dfrac{12}{\sin 40°}$
- $b = \dfrac{12\sin 75°}{\sin 40°} = \dfrac{11.591}{0.6428} = 18.03$ cm

Finding an angle

Use the inverted form, then apply $\sin^{-1}$ .
Worked through — $a = 9$ , $A = 35°$ , $b = 14$ . Find $B$ .
- $\dfrac{\sin B}{14} = \dfrac{\sin 35°}{9}$
- $\sin B = \dfrac{14\sin 35°}{9} = 0.8922$
- $B = \sin^{-1}(0.8922) = 63.1°$

The ambiguous case

This is the trap in the sine rule, and it only arises when you are given two sides and a non-included angle and are solving for an angle.
The issue: $\sin\theta = \sin(180° - \theta)$ . Both $63.1°$ and $116.9°$ have the same sine, so the calculator's answer is only one of two possibilities.
How to check:
- Find the acute angle from $\sin^{-1}$ .
- Calculate the obtuse alternative, $180° - \theta$ .
- Test whether the obtuse option still leaves a positive third angle: does $A + (180° - \theta) < 180°$ ?
- If yes, both triangles are possible and you must give both, or use the context to choose.
- If no, only the acute answer works.
A reliable shortcut: the ambiguous case can only occur when the side opposite the known angle is shorter than the other known side. If the known angle is opposite the longer side, the answer is unique.
Note that the ambiguity never arises when you are finding a side, or when the given angle is already obtuse.

Finding the third angle

The angles of any triangle sum to $180°$ , so once two are known the third is free:

$C = 180° - A - B$

Do this before using the sine rule if it gives you the matching pair you need.

When not to use it

The sine rule needs a complete matching pair. If you know two sides and the angle between them, or all three sides, there is no such pair — use the cosine rule instead.

Tips and Tricks

Put the unknown on top. Use $\frac{a}{\sin A}$ when solving for a side and $\frac{\sin A}{a}$ when solving for an angle. It removes a rearranging step and the error that goes with it.

Check for the ambiguous case every time you find an angle from two sides and a non-included angle. Calculate $180° - \theta$ and test whether it still gives a valid triangle. Missing the second solution is the classic way to lose Merit here.

Degree mode. Check it before you start. An answer of $0.6$ where you expected $60$ is the usual symptom of radian mode.

Find the third angle first when you are given two angles. It often hands you exactly the matching pair the sine rule needs.

Keep full accuracy in intermediate steps. Rounding an angle to the nearest degree part-way through can move a final length by a metre or more in a survey problem.

Sense-check against the triangle. The largest side is always opposite the largest angle. If your answer breaks that, something is wrong.

Worked ExampleThe ambiguous case

In triangle $ABC$ , $a = 8$ cm, $b = 11$ cm and $A = 40°$ . Find all possible values of angle $B$ , and the corresponding values of side $c$ .

Step 1 — Decide which rule applies

We know a side and its opposite angle ( $a = 8$ with $A = 40°$ ) — a complete matching pair — plus one more side. That is the sine rule, in the SSA configuration.

Step 2 — Check whether ambiguity is possible

The known angle $A = 40°$ is opposite side $a = 8$ , and the other known side is $b = 11$ .

Since $8 < 11$ , the side opposite the known angle is the shorter one — so the ambiguous case may arise and both possibilities must be checked.

Step 3 — Set up the sine rule for angle $B$

Put the unknown on top:

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

$\frac{\sin B}{11} = \frac{\sin 40°}{8}$

Step 4 — Solve for $\sin B$

$\sin B = \frac{11 \times \sin 40°}{8} = \frac{11 \times 0.642788}{8}$

$= \frac{7.07067}{8} = 0.883833$

Step 5 — Find both possible angles

The acute solution:

$B = \sin^{-1}(0.883833) = 62.09°$

The obtuse alternative, since $\sin\theta = \sin(180° - \theta)$ :

$B = 180° - 62.09° = 117.91°$

Step 6 — Test whether each gives a valid triangle

Case 1: $B = 62.09°$

$C = 180° - 40° - 62.09° = 77.91°$

Positive ✓ — a valid triangle.

Case 2: $B = 117.91°$

$C = 180° - 40° - 117.91° = 22.09°$

Also positive ✓ — also a valid triangle.

Step 7 — Find $c$ for each case

Case 1, with $C = 77.91°$ :

$\frac{c}{\sin 77.91°} = \frac{8}{\sin 40°}$

$c = \frac{8 \times 0.977866}{0.642788} = \frac{7.82293}{0.642788} = 12.17 \text{ cm}$

Case 2, with $C = 22.09°$ :

$c = \frac{8 \times \sin 22.09°}{\sin 40°} = \frac{8 \times 0.376033}{0.642788} = \frac{3.00826}{0.642788} = 4.68 \text{ cm}$

Step 8 — State both solutions

Step 9 — Sense-check

In each triangle, the largest side is opposite the largest angle:

Triangle 1: largest angle $77.9°$ opposite $c = 12.2$ , the longest side ✓
Triangle 2: largest angle $117.9°$ opposite $b = 11$ , the longest side ✓

Both are internally consistent. In a real context you would use extra information — a sketch, a bearing, or a stated obtuse/acute condition — to decide which triangle is meant.

The sine rule

Covers AS91259

Labelling a triangle

The standard convention makes every formula readable:
- Capital letters ( $A$ , $B$ , $C$ ) label the angles — or the vertices.
- Lower-case letters ( $a$ , $b$ , $c$ ) label the sides.
- Each side is opposite the angle with the same letter. Side $a$ is opposite angle $A$ .

Label your own diagram this way before doing anything else. It converts the formulae from something to memorise into something to read off.

When the sine rule applies

Use the sine rule when you have a matching pair — a side and the angle opposite it.
The two situations:
- Two angles and any side (AAS or ASA)
- Two sides and an angle opposite one of them (SSA) — the ambiguous case

The formula

For finding a side:

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

For finding an angle, turn it upside down so the unknown is on top:

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

Both forms are the same equation. Choosing the one with the unknown in the numerator saves a rearranging step and a chance to slip.
You only ever use two of the three fractions at a time.

Finding a side

Identify the matching pair you know completely.
Write the two fractions, with the unknown side on top.
Multiply across and solve.
Worked through — in triangle $ABC$ , $A = 40°$ , $B = 75°$ and $a = 12$ cm. Find $b$ .
- Matching pair: $a = 12$ with $A = 40°$
- $\dfrac{b}{\sin 75°} = \dfrac{12}{\sin 40°}$
- $b = \dfrac{12\sin 75°}{\sin 40°} = \dfrac{11.591}{0.6428} = 18.03$ cm

Finding an angle

Use the inverted form, then apply $\sin^{-1}$ .
Worked through — $a = 9$ , $A = 35°$ , $b = 14$ . Find $B$ .
- $\dfrac{\sin B}{14} = \dfrac{\sin 35°}{9}$
- $\sin B = \dfrac{14\sin 35°}{9} = 0.8922$
- $B = \sin^{-1}(0.8922) = 63.1°$

The ambiguous case

This is the trap in the sine rule, and it only arises when you are given two sides and a non-included angle and are solving for an angle.
The issue: $\sin\theta = \sin(180° - \theta)$ . Both $63.1°$ and $116.9°$ have the same sine, so the calculator's answer is only one of two possibilities.
How to check:
- Find the acute angle from $\sin^{-1}$ .
- Calculate the obtuse alternative, $180° - \theta$ .
- Test whether the obtuse option still leaves a positive third angle: does $A + (180° - \theta) < 180°$ ?
- If yes, both triangles are possible and you must give both, or use the context to choose.
- If no, only the acute answer works.
A reliable shortcut: the ambiguous case can only occur when the side opposite the known angle is shorter than the other known side. If the known angle is opposite the longer side, the answer is unique.
Note that the ambiguity never arises when you are finding a side, or when the given angle is already obtuse.

Finding the third angle

The angles of any triangle sum to $180°$ , so once two are known the third is free:

$C = 180° - A - B$

Do this before using the sine rule if it gives you the matching pair you need.

When not to use it

The sine rule needs a complete matching pair. If you know two sides and the angle between them, or all three sides, there is no such pair — use the cosine rule instead.

Tips and Tricks

Put the unknown on top. Use $\frac{a}{\sin A}$ when solving for a side and $\frac{\sin A}{a}$ when solving for an angle. It removes a rearranging step and the error that goes with it.

Degree mode. Check it before you start. An answer of $0.6$ where you expected $60$ is the usual symptom of radian mode.

Find the third angle first when you are given two angles. It often hands you exactly the matching pair the sine rule needs.

Keep full accuracy in intermediate steps. Rounding an angle to the nearest degree part-way through can move a final length by a metre or more in a survey problem.

Sense-check against the triangle. The largest side is always opposite the largest angle. If your answer breaks that, something is wrong.

Worked ExampleThe ambiguous case

In triangle $ABC$ , $a = 8$ cm, $b = 11$ cm and $A = 40°$ . Find all possible values of angle $B$ , and the corresponding values of side $c$ .

Step 1 — Decide which rule applies

We know a side and its opposite angle ( $a = 8$ with $A = 40°$ ) — a complete matching pair — plus one more side. That is the sine rule, in the SSA configuration.

Step 2 — Check whether ambiguity is possible

The known angle $A = 40°$ is opposite side $a = 8$ , and the other known side is $b = 11$ .

Since $8 < 11$ , the side opposite the known angle is the shorter one — so the ambiguous case may arise and both possibilities must be checked.

Step 3 — Set up the sine rule for angle $B$

Put the unknown on top:

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

$\frac{\sin B}{11} = \frac{\sin 40°}{8}$

Step 4 — Solve for $\sin B$

$\sin B = \frac{11 \times \sin 40°}{8} = \frac{11 \times 0.642788}{8}$

$= \frac{7.07067}{8} = 0.883833$

Step 5 — Find both possible angles

The acute solution:

$B = \sin^{-1}(0.883833) = 62.09°$

The obtuse alternative, since $\sin\theta = \sin(180° - \theta)$ :

$B = 180° - 62.09° = 117.91°$

Step 6 — Test whether each gives a valid triangle

Case 1: $B = 62.09°$

$C = 180° - 40° - 62.09° = 77.91°$

Positive ✓ — a valid triangle.

Case 2: $B = 117.91°$

$C = 180° - 40° - 117.91° = 22.09°$

Also positive ✓ — also a valid triangle.

Step 7 — Find $c$ for each case

Case 1, with $C = 77.91°$ :

$\frac{c}{\sin 77.91°} = \frac{8}{\sin 40°}$

$c = \frac{8 \times 0.977866}{0.642788} = \frac{7.82293}{0.642788} = 12.17 \text{ cm}$

Case 2, with $C = 22.09°$ :

$c = \frac{8 \times \sin 22.09°}{\sin 40°} = \frac{8 \times 0.376033}{0.642788} = \frac{3.00826}{0.642788} = 4.68 \text{ cm}$

Step 8 — State both solutions

Step 9 — Sense-check

In each triangle, the largest side is opposite the largest angle:

Triangle 1: largest angle $77.9°$ opposite $c = 12.2$ , the longest side ✓
Triangle 2: largest angle $117.9°$ opposite $b = 11$ , the longest side ✓

Both are internally consistent. In a real context you would use extra information — a sketch, a bearing, or a stated obtuse/acute condition — to decide which triangle is meant.

The sine rule

Labelling a triangle

When the sine rule applies

The formula

Finding a side

Finding an angle

The ambiguous case

Finding the third angle

When not to use it

Step 1 — Decide which rule applies

Step 2 — Check whether ambiguity is possible

Step 3 — Set up the sine rule for angle BBB

Step 4 — Solve for sin⁡B\sin BsinB

Step 5 — Find both possible angles

Step 6 — Test whether each gives a valid triangle

Step 7 — Find ccc for each case

Step 8 — State both solutions

Step 9 — Sense-check

The sine rule

Labelling a triangle

When the sine rule applies

The formula

Finding a side

Finding an angle

The ambiguous case

Finding the third angle

When not to use it

Step 1 — Decide which rule applies

Step 2 — Check whether ambiguity is possible

Step 3 — Set up the sine rule for angle BBB

Step 4 — Solve for sin⁡B\sin BsinB

Step 5 — Find both possible angles

Step 6 — Test whether each gives a valid triangle

Step 7 — Find ccc for each case

Step 8 — State both solutions

Step 9 — Sense-check

Step 3 — Set up the sine rule for angle $B$

Step 4 — Solve for $\sin B$

Step 7 — Find $c$ for each case

Step 3 — Set up the sine rule for angle $B$

Step 4 — Solve for $\sin B$

Step 7 — Find $c$ for each case