Cosine rule | Year 11 | Mathematical Methods

01 / LEARN

The idea, step by step

The included angle matters

The cosine rule extends Pythagoras to all triangles. Use the angle between the two known sides. When that angle is 90°, its cosine is zero and the correction term disappears. For an obtuse angle, the cosine is negative, so the opposite side becomes longer.

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

Recovering an angle

Rearrange to $cos C = (a^{2} + b^{2} - c^{2}) / (2 a b)$ before evaluating. The ratio must be between −1 and 1. Check the triangle inequality when three sides are given; impossible side lengths do not become a valid triangle by applying a formula.

A common trap

Using an angle that is not between the two multiplied sides.

Instead: Match the included angle to the pair a,b in the correction term.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

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

$c^{2} = 25 + 49 - 70 cos 60^{\circ}$ .
$c^{2} = 39$ .
Take the positive root because c is a length.

\sqrt{39} cm

Example 2

Worked example 2

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

$cos C = (9 + 16 - 25) / (2 \cdot 3 \cdot 4) = 0$ .
$C = {cos}^{- 1} (0)$ .
The right angle also agrees with Pythagoras.

90°

Check your understanding

What happens when the included angle is 90°?

Show answer

The correction term is zero.

cos 90°=0, so the formula reduces to Pythagoras.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Two sides 8 cm and 8 cm enclose 60°. Find the third side.
3 marks
Two sides 6 cm and 10 cm enclose 120°. 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° 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°. Find the third side.
3 marks
$c^{2} = 64 + 64 - 128 (1 / 2) = 64$ .
8 cm
Two sides 6 cm and 10 cm enclose 120°. Find the third side.
3 marks
$c^{2} = 36 + 100 - 120 (- 1 / 2) = 196$ .
14 cm
Find the angle opposite 7 cm in a 5–6–7 cm triangle to one decimal place.
3 marks
$cos C = (25 + 36 - 49) / 60 = 0.2$ .
$C \approx {78.4630}^{\circ}$ .
78.5°
Find the largest angle in a 13–14–15 cm triangle to one decimal place.
3 marks
$cos C = (169 + 196 - 225) / (2 \cdot 13 \cdot 14) = 5 / 13$ .
$C \approx {67.3801}^{\circ}$ .
67.4°
Two tracks leave a station 45° 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 ↗

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