Topics

The cosine rule and the area of a triangle

Covers AS91259

When the cosine rule applies

Use the cosine rule when there is no matching pair — when the sine rule cannot get started.
The two situations:
- Two sides and the INCLUDED angle (SAS) — the angle between the two known sides
- All three sides (SSS), when you want an angle
The word "included" is the key. If the known angle sits between the two known sides, it is the cosine rule; if it is opposite one of them, it is the sine rule.

Finding a side

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

Each part:
- $a$ — the side you want, opposite the known angle $A$
- $b$ , $c$ — the two sides you know
- $A$ — the included angle, between $b$ and $c$
The structure to notice: it is Pythagoras with a correction term. If $A = 90°$ then $\cos A = 0$ and the formula collapses to $a^2 = b^2 + c^2$ — Pythagoras is the special case.

Worked through — $b = 7$ cm, $c = 10$ cm, $A = 52°$ :
- $a^2 = 49 + 100 - 2(7)(10)\cos 52°$
- $= 149 - 140(0.61566) = 149 - 86.19 = 62.81$
- $a = \sqrt{62.81} = 7.93$ cm
Do not take the square root until the very end, and remember that only the positive root is a length.

Finding an angle

Rearranged to make the angle the subject:

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

The side opposite the angle you want goes in the numerator with the minus sign. Getting that wrong finds a different angle.
Worked through — $a = 9$ , $b = 6$ , $c = 7$ . Find $A$ :
- $\cos A = \dfrac{36 + 49 - 81}{2(6)(7)} = \dfrac{4}{84} = 0.047619$
- $A = \cos^{-1}(0.047619) = 87.3°$
The cosine rule has no ambiguous case. Cosine is negative for obtuse angles and positive for acute ones, so $\cos^{-1}$ returns the single correct answer every time. That is a genuine advantage over the sine rule.
A negative cosine means an obtuse angle — that is information, not an error.

The area of a triangle

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

Each part:
- $a$ , $b$ — two sides
- $C$ — the angle between them
Again the angle must be the included one. The formula is $\frac{1}{2} \times \text{side} \times \text{side} \times \sin(\text{angle between them})$ .
This generalises $\frac{1}{2}\text{base} \times \text{height}$ . The height of the triangle is $b\sin C$ , so the two formulae are the same statement.
Worked through — $a = 8$ cm, $b = 11$ cm, included angle $C = 65°$ :
- Area $= \frac{1}{2}(8)(11)\sin 65° = 44 \times 0.90631 = 39.9$ cm²

Choosing between the rules

Ask what you have been given, in this order:
- Is there a right angle? → SOH CAH TOA and Pythagoras, which are simpler
- Is there a complete matching pair (a side and its opposite angle)? → sine rule
- Otherwise → cosine rule
In a multi-step problem you will often use both. A common pattern is: cosine rule to find one missing side or angle, then sine rule for the rest.
Once you have used the cosine rule to find a side, the sine rule becomes available, because you now have a matching pair.

Multi-step problems

Merit and Excellence questions rarely need one formula.
Work out what you can, then look again. Each new value may unlock a rule that was not available before.
Label everything on the diagram as you find it, so you can see at a glance what is still missing.

Tips and Tricks

"Included" is the deciding word. Two sides with the angle between them means cosine rule. Two sides with an angle opposite one of them means sine rule. Mislabelling which angle you have is the commonest error in the whole standard.

Do not square-root too early. Work out $a^2$ completely, then take the root once at the end. Rooting part of the expression gives a wrong answer that is hard to spot.

A negative cosine is fine. $\cos^{-1}(-0.3) = 107.5°$ tells you the angle is obtuse. Do not discard the minus sign or assume you have made a mistake.

The cosine rule has no ambiguous case, so when a question could go either way, using the cosine rule avoids the problem entirely.

For area, the angle must be between the two sides. $\frac{1}{2}ab\sin C$ needs $C$ enclosed by $a$ and $b$ . If the angle you have is elsewhere, find the included one first.

Sense-check with the largest-side rule and by rough proportions on your diagram. An answer that makes the shortest side face the largest angle is wrong.

Worked ExampleA multi-step problem combining both rules

A triangular paddock has sides $AB = 120$ m and $AC = 95$ m, with the angle at $A$ measuring $73°$ . Find the length of $BC$ , the size of angle $B$ , and the area of the paddock.

Step 1 — Decide which rule to start with

We know two sides ( $AB$ and $AC$ ) and the angle between them ( $A = 73°$ ).

There is no matching pair — we do not know any side together with its opposite angle — so the sine rule cannot start. This is the included angle situation, so the cosine rule applies.

Step 2 — Label using the standard convention

Side $a$ is opposite angle $A$ , so $a = BC$ — the side we want
Side $b$ is opposite $B$ , so $b = AC = 95$ m
Side $c$ is opposite $C$ , so $c = AB = 120$ m

Step 3 — Apply the cosine rule

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

$a^2 = 95^2 + 120^2 - 2(95)(120)\cos 73°$

Work out each piece:

$= 9025 + 14400 - 22800 \times 0.292372$

$= 23425 - 6666.08$

$a^2 = 16758.92$

Step 4 — Take the square root, once, at the end

$a = \sqrt{16758.92} = 129.46$

Sense-check: $73°$ is a fairly wide angle, so the opposite side should be the longest — and $129.5 > 120 > 95$ ✓

Step 5 — Now find angle $B$

We now have a complete matching pair: $a = 129.46$ with $A = 73°$ . That makes the sine rule available, which it was not at the start.

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

$\sin B = \frac{95 \times \sin 73°}{129.46} = \frac{95 \times 0.956305}{129.46}$

$= \frac{90.849}{129.46} = 0.701753$

$B = \sin^{-1}(0.701753) = 44.57°$

Step 6 — Check the ambiguous case

The obtuse alternative would be $180° - 44.57° = 135.43°$ . Testing it:

$C = 180° - 73° - 135.43° = -28.43°$

Negative, so impossible. The obtuse option is rejected and $B = 44.6°$ is the only solution.

Step 7 — Find the area

We know two sides and the included angle, which is exactly what the area formula needs:

$\text{Area} = \frac{1}{2}bc\sin A$

$= \frac{1}{2}(95)(120)\sin 73°$

$= 5700 \times 0.956305 = 5450.9$

Step 8 — Final check

As a cross-check on angle $B$ , find $C$ and confirm the angles sum correctly:

$C = 180° - 73° - 44.57° = 62.43°$

The largest angle is $A = 73°$ opposite the longest side $a = 129.5$ ✓, and the smallest angle $B = 44.6°$ is opposite the shortest side $b = 95$ ✓ Everything is consistent.

The cosine rule and the area of a triangle

Covers AS91259

When the cosine rule applies

Use the cosine rule when there is no matching pair — when the sine rule cannot get started.
The two situations:
- Two sides and the INCLUDED angle (SAS) — the angle between the two known sides
- All three sides (SSS), when you want an angle
The word "included" is the key. If the known angle sits between the two known sides, it is the cosine rule; if it is opposite one of them, it is the sine rule.

Finding a side

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

Each part:
- $a$ — the side you want, opposite the known angle $A$
- $b$ , $c$ — the two sides you know
- $A$ — the included angle, between $b$ and $c$
The structure to notice: it is Pythagoras with a correction term. If $A = 90°$ then $\cos A = 0$ and the formula collapses to $a^2 = b^2 + c^2$ — Pythagoras is the special case.

Worked through — $b = 7$ cm, $c = 10$ cm, $A = 52°$ :
- $a^2 = 49 + 100 - 2(7)(10)\cos 52°$
- $= 149 - 140(0.61566) = 149 - 86.19 = 62.81$
- $a = \sqrt{62.81} = 7.93$ cm
Do not take the square root until the very end, and remember that only the positive root is a length.

Finding an angle

Rearranged to make the angle the subject:

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

The side opposite the angle you want goes in the numerator with the minus sign. Getting that wrong finds a different angle.
Worked through — $a = 9$ , $b = 6$ , $c = 7$ . Find $A$ :
- $\cos A = \dfrac{36 + 49 - 81}{2(6)(7)} = \dfrac{4}{84} = 0.047619$
- $A = \cos^{-1}(0.047619) = 87.3°$
The cosine rule has no ambiguous case. Cosine is negative for obtuse angles and positive for acute ones, so $\cos^{-1}$ returns the single correct answer every time. That is a genuine advantage over the sine rule.
A negative cosine means an obtuse angle — that is information, not an error.

The area of a triangle

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

Each part:
- $a$ , $b$ — two sides
- $C$ — the angle between them
Again the angle must be the included one. The formula is $\frac{1}{2} \times \text{side} \times \text{side} \times \sin(\text{angle between them})$ .
This generalises $\frac{1}{2}\text{base} \times \text{height}$ . The height of the triangle is $b\sin C$ , so the two formulae are the same statement.
Worked through — $a = 8$ cm, $b = 11$ cm, included angle $C = 65°$ :
- Area $= \frac{1}{2}(8)(11)\sin 65° = 44 \times 0.90631 = 39.9$ cm²

Choosing between the rules

Ask what you have been given, in this order:
- Is there a right angle? → SOH CAH TOA and Pythagoras, which are simpler
- Is there a complete matching pair (a side and its opposite angle)? → sine rule
- Otherwise → cosine rule
In a multi-step problem you will often use both. A common pattern is: cosine rule to find one missing side or angle, then sine rule for the rest.
Once you have used the cosine rule to find a side, the sine rule becomes available, because you now have a matching pair.

Multi-step problems

Merit and Excellence questions rarely need one formula.
Work out what you can, then look again. Each new value may unlock a rule that was not available before.
Label everything on the diagram as you find it, so you can see at a glance what is still missing.

Tips and Tricks

Do not square-root too early. Work out $a^2$ completely, then take the root once at the end. Rooting part of the expression gives a wrong answer that is hard to spot.

A negative cosine is fine. $\cos^{-1}(-0.3) = 107.5°$ tells you the angle is obtuse. Do not discard the minus sign or assume you have made a mistake.

The cosine rule has no ambiguous case, so when a question could go either way, using the cosine rule avoids the problem entirely.

For area, the angle must be between the two sides. $\frac{1}{2}ab\sin C$ needs $C$ enclosed by $a$ and $b$ . If the angle you have is elsewhere, find the included one first.

Sense-check with the largest-side rule and by rough proportions on your diagram. An answer that makes the shortest side face the largest angle is wrong.

Worked ExampleA multi-step problem combining both rules

A triangular paddock has sides $AB = 120$ m and $AC = 95$ m, with the angle at $A$ measuring $73°$ . Find the length of $BC$ , the size of angle $B$ , and the area of the paddock.

Step 1 — Decide which rule to start with

We know two sides ( $AB$ and $AC$ ) and the angle between them ( $A = 73°$ ).

There is no matching pair — we do not know any side together with its opposite angle — so the sine rule cannot start. This is the included angle situation, so the cosine rule applies.

Step 2 — Label using the standard convention

Side $a$ is opposite angle $A$ , so $a = BC$ — the side we want
Side $b$ is opposite $B$ , so $b = AC = 95$ m
Side $c$ is opposite $C$ , so $c = AB = 120$ m

Step 3 — Apply the cosine rule

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

$a^2 = 95^2 + 120^2 - 2(95)(120)\cos 73°$

Work out each piece:

$= 9025 + 14400 - 22800 \times 0.292372$

$= 23425 - 6666.08$

$a^2 = 16758.92$

Step 4 — Take the square root, once, at the end

$a = \sqrt{16758.92} = 129.46$

Sense-check: $73°$ is a fairly wide angle, so the opposite side should be the longest — and $129.5 > 120 > 95$ ✓

Step 5 — Now find angle $B$

We now have a complete matching pair: $a = 129.46$ with $A = 73°$ . That makes the sine rule available, which it was not at the start.

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

$\sin B = \frac{95 \times \sin 73°}{129.46} = \frac{95 \times 0.956305}{129.46}$

$= \frac{90.849}{129.46} = 0.701753$

$B = \sin^{-1}(0.701753) = 44.57°$

Step 6 — Check the ambiguous case

The obtuse alternative would be $180° - 44.57° = 135.43°$ . Testing it:

$C = 180° - 73° - 135.43° = -28.43°$

Negative, so impossible. The obtuse option is rejected and $B = 44.6°$ is the only solution.

Step 7 — Find the area

We know two sides and the included angle, which is exactly what the area formula needs:

$\text{Area} = \frac{1}{2}bc\sin A$

$= \frac{1}{2}(95)(120)\sin 73°$

$= 5700 \times 0.956305 = 5450.9$

Step 8 — Final check

As a cross-check on angle $B$ , find $C$ and confirm the angles sum correctly:

$C = 180° - 73° - 44.57° = 62.43°$

The largest angle is $A = 73°$ opposite the longest side $a = 129.5$ ✓, and the smallest angle $B = 44.6°$ is opposite the shortest side $b = 95$ ✓ Everything is consistent.

The cosine rule and the area of a triangle

When the cosine rule applies

Finding a side

Finding an angle

The area of a triangle

Choosing between the rules

Multi-step problems

Step 1 — Decide which rule to start with

Step 2 — Label using the standard convention

Step 3 — Apply the cosine rule

Step 4 — Take the square root, once, at the end

Step 5 — Now find angle BBB

Step 6 — Check the ambiguous case

Step 7 — Find the area

Step 8 — Final check

The cosine rule and the area of a triangle

When the cosine rule applies

Finding a side

Finding an angle

The area of a triangle

Choosing between the rules

Multi-step problems

Step 1 — Decide which rule to start with

Step 2 — Label using the standard convention

Step 3 — Apply the cosine rule

Step 4 — Take the square root, once, at the end

Step 5 — Now find angle BBB

Step 6 — Check the ambiguous case

Step 7 — Find the area

Step 8 — Final check

Step 5 — Now find angle $B$

Step 5 — Now find angle $B$