The cosine rule
I can derive the formula for the cosine rule.
The cosine rule
I can derive the formula for the cosine rule.
These resources will be removed by end of Summer Term 2025.
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.
Licence
This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Lesson video
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Starter quiz
6 Questions
The sine of an angle is the -
$$y$$-coordinate of P on the triangle formed inside the unit circle.
The cosine of an angle is the -
$$x$$-coordinate of P on the triangle formed inside the unit cirlce.
The tangent is the -
line that intersects the circle exactly once.
1 -
$$\cos(0°)$$
$$\frac{\sqrt{3}}{2}$$ -
$$\cos(30°)$$
$$\frac{\sqrt{2}}{2}$$ -
$$\cos(45°)$$
0.5 -
$$\cos(60°)$$
0 -
$$\cos(90°)$$
-0.174 (3 s.f) -
$$\cos(100°)$$
Exit quiz
6 Questions
Pythagoras' theorem -
$$a^2+b^2=c^2$$
The sine rule -
$$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
The cosine rule -
$$a^2 = b^2+c^2-2bc\cos(A)$$
Area of a triangle -
$$\frac{1}{2}ab\sin(C)$$
10.6 cm -
$$a=8$$ cm, $$B=36$$° and $$c=16$$ cm
11.8 cm -
$$a=16$$ cm, $$B=36$$° and $$c=20$$ cm
4.9 cm -
$$a=8$$ cm, $$B=36$$° and $$c=8$$ cm
12.2 cm -
$$a=8$$ cm, $$B=72$$° and $$c=12$$ cm
46.57° -
$$a=8$$ cm, $$b=12$$ cm and $$c=16$$ cm
36.87° -
$$a=16$$ cm, $$b=12$$ cm and $$c=20$$ cm
97.18° -
$$a=8$$ cm, $$b=12$$ cm and $$c=8$$ cm
70.53° -
$$a=8$$ cm, $$b=12$$ cm and $$c=12$$ cm