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Lesson details

Key learning points

There are multiple ways to prove the cosine rule
When deriving (or proving) it is sometimes useful to derive other information

Keywords

Cosine rule - The cosine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.

Common misconception

Pupils may not see the need for this rule as they have the sine rule.

The sine rule can be used in many cases but it is not always the most efficient. There will be problems where it will be much faster to use the cosine rule.

To help you plan your year 11 maths lesson on: The cosine rule, download all teaching resources for free and adapt to suit your pupils' needs...

Pupils may benefit from having an annotated triangle displayed on a separate board along with the cosine rule so this can be easily referenced.

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Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.

Which of the following are equivalent to

\cos (θ °) = \frac{adj}{hyp}

?

Correct answer:

hyp \times \cos (θ °) = adj

Correct answer:

hyp = \frac{adj}{\cos (θ °)}

\cos (θ °) = \frac{hyp}{adj}

Q2.

Match the trigonometric ratio with the correct definition.

Correct Answer:The sine of an angle is the ,

y

-coordinate of P on the triangle formed inside the unit circle.

$y$ -coordinate of P on the triangle formed inside the unit circle.

Correct Answer:The cosine of an angle is the ,

x

-coordinate of P on the triangle formed inside the unit cirlce.

$x$ -coordinate of P on the triangle formed inside the unit cirlce.

Correct Answer:The tangent is the ,line that intersects the circle exactly once.

line that intersects the circle exactly once.

Q3.

Match the following values.

Correct Answer:1,

\cos (0 °)

$\cos (0 °)$

Correct Answer:

\frac{\sqrt{3}}{2}

,

\cos (30 °)

$\cos (30 °)$

Correct Answer:

\frac{\sqrt{2}}{2}

,

\cos (45 °)

$\cos (45 °)$

Correct Answer:0.5,

\cos (60 °)

$\cos (60 °)$

Correct Answer:0,

\cos (90 °)

$\cos (90 °)$

Correct Answer:-0.174 (3 s.f),

\cos (100 °)

$\cos (100 °)$

Q4.

A right-angled triangle has a height of 15 cm and a hypotenuse of 17 cm. Work out the area.

Correct answer: 60 cm

^{2}

120 cm

^{2}

30 cm

^{2}

240 cm

^{2}

Q5.

An isosceles triangle has a base of 16 cm and base angles of 53.97° (2 d.p). The area of this triangle, given to 2 significant figures, is cm

^{2}

.

Correct Answer: 88

Q6.

Six equilateral triangles are put together to make a regular hexagon. The length of each triangle is 10 cm. Work out the exact area of the regular hexagon.

75 \sqrt{3}

cm

^{2}

Correct answer:

150 \sqrt{3}

cm

^{2}

25 \sqrt{3}

cm

^{2}

500 \sqrt{3}

cm

^{2}

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6 Questions

Q1.

Match the correct rules, theorem or formulae.

Correct Answer:Pythagoras' theorem,

a^{2} + b^{2} = c^{2}

$a^{2} + b^{2} = c^{2}$

Correct Answer:The sine rule,

\frac{a}{\sin (A)} = \frac{b}{\sin (B)} = \frac{c}{\sin (C)}

$\frac{a}{\sin (A)} = \frac{b}{\sin (B)} = \frac{c}{\sin (C)}$

Correct Answer:The cosine rule,

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

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Correct Answer:Area of a triangle,

\frac{1}{2} a b \sin (C)

$\frac{1}{2} a b \sin (C)$

Q2.

Which of the following are the cosine rule?

Correct answer:

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

Correct answer:

b^{2} = a^{2} + c^{2} - 2 a c \cos (B)

Correct answer:

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

c^{2} = a^{2} + b^{2} - 2 a c \cos (B)

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

Q3.

A triangle has adjacent lengths of 16 cm and 20 cm with an angle in between of 55°. Work out the missing length to 1 decimal place.

Correct answer: 17.0 cm

8.5 cm

34.0 cm

12.3 cm

Q4.

This proof has been mixed up. Write each line in order, starting with

p^{2} + (b - r)^{2} = a^{2}

.

1 -

p^{2} + (b - r)^{2} = a^{2}

2 -

p^{2} + b^{2} - 2 b r + r^{2} = a^{2}

3 -

p^{2} = c^{2} - r^{2}

4 -

c^{2} - r^{2} + b^{2} - 2 b r + r^{2} = a^{2}

5 -

a^{2} = b^{2} + c^{2} - 2 b r

6 - Given

\cos (A) = \frac{r}{c}

and

r = c \cos (A)

7 -

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

Q5.

Given a triangle with lengths labelled

a

,

b

and

c

, match up the length

b

with dimensions of the triangle and the angle

B

.

Correct Answer:10.6 cm,

a = 8

cm,

B = 36

° and

c = 16

cm

$a = 8$ cm, $B = 36$ ° and $c = 16$ cm

Correct Answer:11.8 cm,

a = 16

cm,

B = 36

° and

c = 20

cm

$a = 16$ cm, $B = 36$ ° and $c = 20$ cm

Correct Answer:4.9 cm,

a = 8

cm,

B = 36

° and

c = 8

cm

$a = 8$ cm, $B = 36$ ° and $c = 8$ cm

Correct Answer:12.2 cm,

a = 8

cm,

B = 72

° and

c = 12

cm

$a = 8$ cm, $B = 72$ ° and $c = 12$ cm

Q6.

Given a triangle with lengths labelled

a

,

b

and

c

, match up the angle

B

with the dimensions of the triangle.

Correct Answer:46.57°,

a = 8

cm,

b = 12

cm and

c = 16

cm

$a = 8$ cm, $b = 12$ cm and $c = 16$ cm

Correct Answer:36.87°,

a = 16

cm,

b = 12

cm and

c = 20

cm

$a = 16$ cm, $b = 12$ cm and $c = 20$ cm

Correct Answer:97.18°,

a = 8

cm,

b = 12

cm and

c = 8

cm

$a = 8$ cm, $b = 12$ cm and $c = 8$ cm

Correct Answer:70.53°,

a = 8

cm,

b = 12

cm and

c = 12

cm

$a = 8$ cm, $b = 12$ cm and $c = 12$ cm

The cosine rule

The cosine rule

These resources will be removed by end of Summer Term 2025.

Slide deck

Lesson details

Key learning points

Keywords

Common misconception

How to plan a lesson using our resources

Licence

Lesson video

Worksheet

Starter quiz

6 Questions

Exit quiz

6 Questions