1. The Cosine Rule

2. Starter Work out the value of the missing angle, θ. Sinθ 20 Sin 71 22= θ
22mm
Sin 71 22
Sinθ = x 20
71°
Sinθ = …
20mm
Sin-1 θ =
59.3° (1dp)

3. The Cosine Rule
Last lesson we saw the Sine Rule, which linked opposite sides and angles in any triangle
This involved the use of Sine
This lesson we will learn to use the Cosine Rule, which also allows us to find a missing side or angle…

4. The Cosine Rule Learning Objectives
All will be able to use the cosine rule to calculate missing sides in a triangle (Grade A/A*)
Most will also be able to use this rule to calculate missing angles (Grade A*)
Some will be able to use this in conjunction with the Sine rule and others in order to answer more problems! (Grade A*)

5. The Cosine Rule
The Cosine Rule states that in any triangle, labelled as to the right;
a2 = b2 + c2 - 2bc cosA
b2 = a2 + c2 – 2ac cosB
c2 = a2 + b2 – 2ab cos C B c a A b C

6. The Cosine Rule Find the missing side in the triangle opposite.
a2 = b2 + c2 – 2bc cosA
a2 = – (2x7x9)cos 81
a2 = – 126cos81
a2 = 130 – 19.71…
a = 10.5cm (1dp)
a2 = b2 + c2 - 2bc cosA b a
7cm ?
81º A
9cm c √

7. The Cosine Rule Find the missing side in the triangle opposite.
a2 = b2 + c2 – 2bc cosA
a2 = – (2x10x18)cos 32
a2 = – 360cos32
a2 = 424 – …
a = 10.9cm (1dp)
a2 = b2 + c2 - 2bc cosA A b
10m
32º
18m c ? √ a

8. The Cosine Rule Find the missing angle in the triangle opposite.
a2 = b2 + c2 – 2bc cosA
112 = – (2x13x12)cosA
121 = – 312cosA
121 = 313 – 312cosA
-192 = - 312cosA
0.615… = cos A
A = 52.02º (2dp)
a2 = b2 + c2 - 2bc cosA A ?
12m c b
13m
- 313
÷ -312
11m
Cos-1 a

9. The Cosine Rule Find the missing angle in the triangle opposite.
a2 = b2 + c2 – 2bc cosA
52 = – (2x15x11)cosA
25 = – 330cosA
25 = 346 – 330cosA
-321 = - 330cosA
0.972… = cos A
A = 13.41º (2dp)
a2 = b2 + c2 - 2bc cosA c
15m
- 346 a 5m
÷ -330 ? A
11m
Cos-1 b

10. Plenary Find the missing side in the triangle opposite.
a2 = b2 + c2 – 2bc cosA
a2 = – (2x7x14)cos90
a2 = – 196cos90
a2 = 245 – 196cos90
a2 = 245
a = 15.6 (1dp)
a2 = b2 + c2 – 2bc cosA
(Using the cosine rule)
15.6m a b ? 7m
196Cos90 = 0 A
14m √ c
If the angle used is 90, then the part with Cos in will be 0…
So a2 = b2 + c2  PYTHAGORAS!

11. Plenary Learning Objectives
All will be able to use the cosine rule to calculate missing sides in a triangle (Grade A/A*)
Most will also be able to use this rule to calculate missing angles (Grade A*)
Some will be able to use this in conjunction with the Sine rule and others in order to answer more problems! (Grade A*)

12. Summary We have learnt how to use the Cosine Rule
We have seen that it works in any triangle
We have used it to find both missing sides and angles