Cosine rule | Year 11 | Mathematical Methods

01 / LEARN

Understand the idea.

The included angle matters

The cosine rule uses two sides and the angle between them to find the opposite side. Its correction term accounts for the angle: a right angle gives Pythagoras, while an obtuse angle makes the opposite side longer than the Pythagorean comparison.

c^{2} = a^{2} + b^{2} - 2 a b cos C

Recovering an angle

With three sides known, isolate $cos C$ before applying inverse cosine. The side squared with the minus sign is the side opposite the required angle. A triangle angle lies strictly between $0^{\circ}$ and $180^{\circ}$ .

First check that the sum of any two sides exceeds the third. Invalid side data cannot form a triangle, even if you can enter them into a formula. For valid data the cosine ratio lies between $- 1$ and $1$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Two sides are 5 cm and 7 cm with included angle 60°. Find the third side exactly.

Worked solution

1
Use the angle between the known sides.
$c^{2} = 5^{2} + 7^{2} - 2 (5) (7) cos 60^{\circ}$
The unknown side lies opposite the included angle.
2
Evaluate the cosine correction.
$c^{2} = 25 + 49 - 70 (\frac{1}{2}) = 39$
The exact cosine is $\frac{1}{2}$ , so no rounding is needed.
3
Take the positive square root.
$c = \sqrt{39} cm$
A side length is positive; the negative square root is not a geometric solution.

Answer

$\sqrt{39} cm$

EXAMPLE 02

Worked example 2

Find the angle opposite side 5 in a triangle with sides 3,4,5.

Worked solution

1
Place the opposite side in the angle formula.
$cos C = \frac{3^{2} + 4^{2} - 5^{2}}{2 (3) (4)}$
The required angle faces side 5, so its square is subtracted.
2
Simplify the ratio.
$cos C = \frac{9 + 16 - 25}{24} = 0$
The two smaller squares sum to the largest square.
3
Find the triangle angle.
$C = {cos}^{- 1} (0) = 90^{\circ}$
Within the triangle angle interval, cosine is zero only at $90^{\circ}$ .

Answer

90°

03 / YOUR TURN

Try it yourself.

Check your understanding

What happens when the included angle is 90°?

Show answer & reasoning

The correction term is zero.

Since $cos 90^{\circ} = 0$ , the term $2 a b cos C$ vanishes. The equation becomes $c^{2} = a^{2} + b^{2}$ , exactly Pythagoras.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Two sides 8 cm and 8 cm enclose $60^{\circ}$ . Find the third side.
3 marks
Two sides 6 cm and 10 cm enclose $120^{\circ}$ . Find the third side.
3 marks
Find the angle opposite 7 cm in a 5–6–7 cm triangle to one decimal place.
3 marks
Find the largest angle in a 13–14–15 cm triangle to one decimal place.
3 marks
Two tracks leave a station $45^{\circ}$ apart. Trains are 4 km and 7 km from it. Find their separation to two decimal places.
3 marks
Can sides 2,3,6 cm form a triangle? Justify.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two sides 8 cm and 8 cm enclose $60^{\circ}$ . Find the third side.
3 marks
$c^{2} = 64 + 64 - 128 (\frac{1}{2}) = 64$ .
8 cm
Two sides 6 cm and 10 cm enclose $120^{\circ}$ . Find the third side.
3 marks
$c^{2} = 36 + 100 - 120 (- \frac{1}{2}) = 196$ .
14 cm
Find the angle opposite 7 cm in a 5–6–7 cm triangle to one decimal place.
3 marks
$cos C = \frac{25 + 36 - 49}{60} = 0.2$ .
$C \approx {78.4630}^{\circ}$ .
${78.5}^{\circ}$
Find the largest angle in a 13–14–15 cm triangle to one decimal place.
3 marks
$cos C = \frac{169 + 196 - 225}{2 \cdot 13 \cdot 14} = \frac{5}{13}$ .
$C \approx {67.3801}^{\circ}$ .
${67.4}^{\circ}$
Two tracks leave a station $45^{\circ}$ apart. Trains are 4 km and 7 km from it. Find their separation to two decimal places.
3 marks
$d = \sqrt{65 - 28 \sqrt{2}} \approx 5.04004$ .
5.04 km
Can sides 2,3,6 cm form a triangle? Justify.
3 marks
$2 + 3 < 6$ , violating the triangle inequality.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

In this lesson

Understand the idea.

The included angle matters

Recovering an angle

See the method in action.

Worked example 1

Use the angle between the known sides.

Evaluate the cosine correction.

Take the positive square root.

Worked example 2

Place the opposite side in the angle formula.

Simplify the ratio.

Find the triangle angle.

Try it yourself.

Check your understanding

Keep practising