1. The Cosine Rule. A B C a b c
a2 = b2 + c2
-2bccosAo

2. A B C a b c h D x
c-x
c-x B C a h D
Apply Pythagoras to triangle CDB.
a2 = h2 +
(c - x) 2
a2 = h2 + c2
-2cx
+ x2
Square out the bracket.
a2 = b2 + c2
-2cx b2
What does h2 and x2 make?
a2 = b2 + c2
-2cbcosAo
What does the cosine of Ao equal?
We now have: x
cos Ao =
Make x the subject: b
a2 = b2 + c2
-2bccosAo
Substitute into the formula:
x =
bcosAo
The Cosine Rule.

3. When To Use The Cosine Rule.
The Cosine Rule can be used to find a third side of a triangle if you have the other two sides and the angle between them.
All the triangles below are suitable for use with the Cosine Rule: 6 10
65o L
89o
13.8
6.2 W
147o 8 11 M
Note the pattern of sides and angle.

4. Using The Cosine Rule. Example 1.
Find the unknown side in the triangle below: L 5m
12m
43o
Identify sides a,b,c and angle Ao
Write down the Cosine Rule.
c = 12
Ao =
43o
a = L
b = 5
a2 = b2 + c2
-2bccosAo
Substitute values and find a2.
a2 = 52 +
122
- 2 x 5 x 12 cos 43o
a2 = -
(120 x
0.731 )
a2 =
81.28
Square root to find “a”.
a = 9.02m

5. Example 2.
Find the length of side M.
137o
17.5 m
12.2 m M
Identify the sides and angle.
a = M
b = 12.2
C = 17.5
Ao = 137o
Write down Cosine Rule and substitute values.
a2 = b2 + c2
-2bccosAo
a2 = – ( 2 x 12.2 x 17.5 x cos 137o )
a2 = – ( 427 x – )
Notice the two negative signs.
a2 =
a2 =
a = 27.7m

6. What Goes In The Box ? 1.
Find the length of the unknown side in the triangles below:
(1)
78o
43cm
31cm L
G = 12.4cm
(3)
110o
6.3cm
8.7cm G
L = 47.5cm
(2) 8m
5.2m
38o M
M =5.05m

7. Finding Angles Using The Cosine Rule.
Consider the Cosine Rule again:
a2 = b2 + c2
-2bccosAo
We are going to change the subject of the formula to cos Ao
Turn the formula around:
b2 + c2 – 2bc cos Ao = a2
-2bc cos Ao = a2 – b2 – c2
Take b2 and c2 across.
Divide by – 2 bc.
Divide top and bottom by -1
You now have a formula for finding an angle if you know all three sides of the triangle.

8. Finding An Angle. Example 1
Use the formula for Cos Ao to calculate the unknown angle xo below: xo
16cm
9cm
11cm
Ao = xo
a = 11
b = 9
c = 16
Write down the formula for cos Ao
Identify Ao and a , b and c.
Substitute values into the formula.
Cos Ao =
0.75
Calculate cos Ao .
Ao = 41.4o
Use cos to find Ao

9. Example 2.
Find the unknown angle in the triangle below:
26cm
15cm
13cm yo
Write down the formula.
Identify the sides and angle.
Substitute into the formula.
Ao = yo
a = 26
b = 15
c = 13
Find the value of cosAo
The negative tells you the angle is obtuse.
cosAo =
Ao =
136.3o

10. What Goes In The Box ? 2
Calculate the unknown angles in the triangles below:
(1)
10m 7m 5m ao
(3) co
27cm
14cm
16cm
ao =111.8o
(2)
12.7cm
7.9cm
8.3cm bo
bo = 37.3o
co =128.2o