Cosine rule | Year 10 | Ascent Library

01 / LEARN

The idea, step by step

The included angle matters

Use the cosine rule when you know two sides and the angle between them. The unknown side is opposite that angle. For example, if A lies between sides b and c, the opposite side satisfies a² = b² + c² − 2bc cos A.

For a right angle the cosine term is zero, so this becomes Pythagoras. An obtuse included angle has negative cosine and makes the opposite side larger.

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

Reverse the rule to find an angle

With all three sides known, rearrange first: cos A = (b² + c² − a²)/(2bc). Use inverse cosine only after the entire fraction has been evaluated.

The longest side must face the largest angle. A useful check is whether the answer is acute, right or obtuse. The three sides must also satisfy the triangle inequality.

A common trap

Subtracting the cosine after taking a square root.

Instead: Calculate the complete expression for the square of the side, then take its positive square root.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two sides and the included angle

Two sides are 7 cm and 10 cm, with included angle 60°. Find the third side to 0.1 cm.

$c^{2} = 7^{2} + 10^{2} - 2 (7) (10) cos 60^{\circ} = 79$ .
$c = \sqrt{79}$ .

8.9 cm

Example 2

Find an angle

A triangle has side lengths 5 cm, 7 cm and 8 cm. Find the angle opposite 8 cm to 0.1°.

$cos C = (5^{2} + 7^{2} - 8^{2}) / (2 \cdot 5 \cdot 7) = 1 / 7$ .
$C = {cos}^{- 1} (1 / 7)$ .

81.8°

Check your understanding

Sides 3 cm and 4 cm include a right angle. What side does the cosine rule give?

Show answer

5 cm

$cos 90^{\circ} = 0$ , so $c = \sqrt{3^{2} + 4^{2}} = 5$ cm.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Sides 3 m and 5 m include 60°. Find the opposite side exactly.
2 marks
A triangle has sides 7 cm, 8 cm and 9 cm. Find the angle opposite 9 cm to 0.1°.
3 marks
Sides 10 m and 12 m include 120°. Find the third side to 0.1 m.
3 marks
A triangle has sides 8 cm, 15 cm and 17 cm. Use the cosine rule to classify the angle opposite 17 cm.
3 marks
In triangle ABC, b = 8 cm, c = 11 cm and A = 40°. Find a to 0.1 cm.
3 marks
You know all three sides but no angles. Which rule gives a starting angle, and why is the sine rule not a starting method?
2 marks

Read the worked solutions

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

Sides 3 m and 5 m include 60°. Find the opposite side exactly.
2 marks
$c^{2} = 3^{2} + 5^{2} - 2 (3) (5) cos 60^{\circ} = 19$ .
$c = \sqrt{19}$ m.
$\sqrt{19}$ m
A triangle has sides 7 cm, 8 cm and 9 cm. Find the angle opposite 9 cm to 0.1°.
3 marks
$cos C = (49 + 64 - 81) / 112 = 2 / 7$ .
$C = {cos}^{- 1} (2 / 7)$ .
73.4°
Sides 10 m and 12 m include 120°. Find the third side to 0.1 m.
3 marks
$c^{2} = 100 + 144 - 240 cos 120^{\circ} = 364$ .
$c = \sqrt{364}$ .
19.1 m
A triangle has sides 8 cm, 15 cm and 17 cm. Use the cosine rule to classify the angle opposite 17 cm.
3 marks
$cos C = (64 + 225 - 289) / (2 \cdot 8 \cdot 15) = 0$ .
$C = 90^{\circ}$ .
90°; it is a right angle.
In triangle ABC, b = 8 cm, c = 11 cm and A = 40°. Find a to 0.1 cm.
3 marks
$a^{2} = 8^{2} + 11^{2} - 2 (8) (11) cos 40^{\circ}$ .
Take the positive square root.
7.1 cm
You know all three sides but no angles. Which rule gives a starting angle, and why is the sine rule not a starting method?
2 marks
The cosine rule can use three side lengths.
The sine rule needs at least one complete opposite pair.
Use the cosine rule; no opposite side–angle pair is known for the sine rule.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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