Cosine rule | Year 10 | Ascent Library

01 / LEARN

Understand the idea.

The included angle matters

The cosine rule finds a third side from two sides and their included angle. The angle between the known sides must be opposite the unknown side; mark this relationship before substituting.

When the included angle is $90^{\circ}$ , its cosine is zero and the rule becomes Pythagoras. For an obtuse angle, the cosine is negative, so subtracting the cosine term increases the opposite side's squared length.

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

Reverse the rule to find an angle

With three sides known, isolate the cosine of the required angle before applying inverse cosine. Keep the whole numerator grouped and divide by twice the product of the two sides enclosing that angle.

The sides must satisfy the triangle inequality. The longest side must face the largest angle, and the calculated cosine must lie between $- 1$ and $1$ . These checks can reveal mismatched labels or arithmetic errors.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Choose the side opposite the included angle.
$c^{2} = 7^{2} + 10^{2} - 2 (7) (10) cos 60^{\circ}$
The two given sides meet at the given $60^{\circ}$ angle.
2
Evaluate the complete squared length.
$c^{2} = 49 + 100 - 140 (\frac{1}{2}) = 79$
The cosine term belongs inside the expression for $c^{2}$ .
3
Take the positive root and round.
$c = \sqrt{79} \approx 8.9 cm$
A side length is positive; the root is taken after all squared-length operations.

Answer

8.9 cm

EXAMPLE 02

Find an angle

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

Worked solution

1
Rearrange for the angle opposite the longest side.
$cos C = \frac{5^{2} + 7^{2} - 8^{2}}{2 (5) (7)}$
The side opposite $C$ is $8 cm$ .
2
Simplify the ratio.
$cos C = \frac{25 + 49 - 64}{70} = \frac{1}{7}$
The numerator and denominator must both be completed before inversion.
3
Apply inverse cosine in degrees.
$C = {cos}^{- 1} (\frac{1}{7}) \approx {81.8}^{\circ}$
Cosine determines a unique triangle angle between $0^{\circ}$ and $180^{\circ}$ .

Answer

81.8°

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

5 cm

Because $cos 90^{\circ} = 0$ , the cosine rule reduces to $c^{2} = 3^{2} + 4^{2} = 25$ , giving $c = 5 cm$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Sides 3 m and 5 m include $60^{\circ}$ . 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 ${0.1}^{\circ}$ .
3 marks
Sides 10 m and 12 m include $120^{\circ}$ . 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 $A B C$ , $b = 8$ cm, $c = 11$ cm and $A = 40^{\circ}$ . 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 $60^{\circ}$ . 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 ${0.1}^{\circ}$ .
3 marks
$cos C = \frac{49 + 64 - 81}{112} = \frac{2}{7}$ .
$C = {cos}^{- 1} (\frac{2}{7})$ .
${73.4}^{\circ}$
Sides 10 m and 12 m include $120^{\circ}$ . 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
$cos C = \frac{64 + 225 - 289}{2 \cdot 8 \cdot 15} = 0$ .
$C = 90^{\circ}$ .
$90^{\circ}$ ; it is a right angle.
In triangle $A B C$ , $b = 8$ cm, $c = 11$ cm and $A = 40^{\circ}$ . 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 ↗

In this lesson

Understand the idea.

The included angle matters

Reverse the rule to find an angle

See the method in action.

Two sides and the included angle

Choose the side opposite the included angle.

Evaluate the complete squared length.

Take the positive root and round.

Find an angle

Rearrange for the angle opposite the longest side.

Simplify the ratio.

Apply inverse cosine in degrees.

Try it yourself.

Check your understanding

Keep practising