“Teach A Level Maths”
Vol. 1: AS Core Modules
36: The Cosine Rule
The Cosine Rule
Module C2
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The Cosine Rule
The cosine rule is used to find sides and angles of
a scalene triangle when
•
2 sides and the angle formed by them are
known, or
•
all 3 sides are known
In both these cases, we don’t know a pair of side
and opposite angle so the sine rule cannot be used.
We will now prove the cosine rule but you do not
need to learn the proof.
The Cosine Rule
Proof of the Cosine Rule
In the triangle ABC, draw the perpendicular, h,
C
from C to AB.
Let AN = x. Then, NB = c - x.
b
x
a
From triangle ANC, cos A 
h
 x  b cos A - - - - (1)
b
x
A
Using Pythagoras’ theorem:
In triangle ANC, h2  b 2 - x 2
In triangle BNC, h2  a - (c - x )
2
So,
c-x
c
N
B
2
b 2 - x 2  a 2 - (c - x ) 2 - - - - (2)
The Cosine Rule
Proof of the Cosine Rule
We have
b 2 - x 2  a 2 - (c - x ) 2 - - - - (2)
Simplifying:
b 2 - x 2  a 2 - (c 2 - 2cx  x 2 )
2
2
2
2
2
b
x

a
c

2
cx
x

2
2
2
b

a
c
 2cx

Substituting for x from equation (1), ( x  b cos A )

Rearranging:
2
2
b 2  a 2 - c 2  2cb cos A
b  c - 2bc cos A  a

2
a 2  b 2  c 2 - 2bc cos A
The Cosine Rule
The Cosine Rule for triangle ABC
a  b  c - 2bc cos A
2
2
2
•
We use this arrangement when 2 sides and the
angle formed by them are known.
•
The letters can be switched to find any side
provided it is opposite the given angle.
The Cosine Rule
The Cosine Rule for triangle ABC
a  b  c - 2bc cos A
2
2
2
•
We use this arrangement when 2 sides and the
angle formed by them are known.
•
The letters can be switched to find any side
provided it is opposite the given angle.
•
If we want to find an angle, we use the sine rule
after we have used the cosine rule.
The Cosine Rule
A
e.g. Find side c and angle B in
the triangle ABC
Solution:
Use the Cosine rule
C
b
15
30 
c 2  b 2  a 2 - 2ba cos C

c
19 a
c 2  15 2  19 2 - 2(15)(19) cos 30
c Do
9 61
the
( 3whole
s.f.) calculation
 Tip:
in sin
oneB go sin
onCyour calculator.
15 sin 30
The Sine rule:
Tip: Leave
the answer
sin on
B
 avoids

It
errors!
b
c as it will be 9  61
your calculator

B  51 3 ( 3 s.f.)
needed to find angle B
B
The Cosine Rule
Exercise
1. Find p in the triangle PQR
Q
p
6
120
P
Solution:
7
R
p 2  q 2  r 2 - 2qr cos P


p 2  7 2  6 2 - 2(7)( 6) cos 120
p  11 3 ( 3 s.f.)
The Cosine Rule
The 2nd form of the Cosine Rule
We know that
Rearranging,
a 2  b 2  c 2 - 2bc cos A
2bc cos A  b 2  c 2 - a 2

b2  c2 - a 2
cos A 
2bc
The minus sign . . .
. . . belongs to the side
opposite the angle we are finding
We use this form to find any angle of a triangle
when we know all 3 sides.
The Cosine Rule
The Cosine Rule
8
X
e.g. 1 Find angle X in
triangle XYZ
Solution: Use the Cosine Rule
y z -x
cos X 
2 yz
2
2
82  62 - 42
 cos X 
2(8)( 6)

cos X  0  875

X  29 0 
4
6
2
Z
Y
The Cosine Rule
C
e.g. 2 Find all the angles
in triangle ABC
Solution: Let’s find A first
9
A
6
5
B
b2  c2 - a 2
92  52 - 62
 cos A 
cos A 
2bc
2(9)( 5)

A

38
9

cos
A

0

7778

We can now use the Cosine rule again or switch to
the Sine rule. If we use the Sine rule, we must
avoid the largest angle ( opposite the longest side )
as we don’t know whether it is less than or greater
than 90.
The Cosine Rule
C
9
A
A  38 9 
6
5
B
EITHER: Using the Cosine rule for B or C:
e.g.
52  62 - 92
cos B 
 cos B  - 0  333  B  109 5 
2(5)( 6)
Then
C  180 - 109  5 - 38  9  31  6 
OR: Using the Sine rule for C :
5 sin 38  9
sin C sin A
 sin C 

6
c
a
Then
 C  31 6 
B  180 - 31  6 - 38  9  109  5 
The Cosine Rule
SUMMARY
 The Cosine Rule
• In a triangle that isn’t right angled, if we
know 2 sides and the angle formed by the 2
sides, we use
a 2  b 2  c 2 - 2bc cos A
to find the 3rd side.
•
If we know 3 sides, we use
b2  c2 - a 2
cos A 
2bc
to find any angle.
The Cosine Rule
Exercise
1. Find all the angles in the triangle XYZ giving
the answers to 1 decimal place.
X
9
4
Y
Z
7
Solution: Use the Cosine rule for any angle. e.g.
y2  z2 - x2
42  92 - 72
cos X 
 cos X 
 X  48 2 
2 yz
2(4)( 9)
72  92 - 42
x2  z2 - y2
 cos Y 
cos Y 
 Y  25 2 
2(7)( 9)
2 xz
Z  180 - 25  2 - 48  2  106  6  (1 d. p.)
The Cosine Rule
The Cosine Rule
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
The Cosine Rule
The cosine rule is used to find sides and angles of
a scalene triangle when
•
2 sides and the angle formed by them are
known, or
•
all 3 sides are known
In both these cases, we don’t know a side and its
opposite angle so the sine rule cannot be used.
The Cosine Rule
The Cosine Rule for triangle ABC
a  b  c - 2bc cos A
2
2
2
•
We use this arrangement when 2 sides and the
angle formed by them are known.
•
The letters can be switched to find any side
provided it is opposite the given angle.
•
If we want to find an angle, we use the sine rule
after we have used the cosine rule.
The Cosine Rule
e.g. Find side c and angle B in
the triangle ABC
Solution: Use the Cosine rule
c 2  b 2  a 2 - 2ba cos C
C
A
b
15
c
30 
2
2
2
 c  15  19 - 2(15)(19) cos 30
19 a
Tip: Do the whole calculation in one go on your
calculator and leave your answer so it can be used to
find B.
 c  9 61 ( 3 s.f.)
sin B sin C

The Sine rule:

b
c

15 sin 30
sin B 
9  61
B  51 3 ( 3 s.f.)
B
The Cosine Rule
The 2nd form of the Cosine Rule
We know that
Rearranging,
a 2  b 2  c 2 - 2bc cos A
2bc cos A  b 2  c 2 - a 2
b2  c2 - a 2
 cos A 
2bc
The minus sign goes with
the side opposite the
angle we are finding.
We use this form to find any angle of a triangle
when we know all 3 sides.
The Cosine Rule
C
e.g. 2 Find all the angles
in triangle ABC
Solution: Let’s find A first
9
A
6
5
B
b2  c2 - a 2
92  52 - 62
 cos A 
cos A 
2bc
2(9)( 5)

A

38
9

cos
A

0

7778

We can now use the Cosine rule again or switch to
the Sine rule. If we use the Sine rule, we must
avoid the largest angle ( opposite the longest side )
as we don’t know whether it is less than or greater
than 90.
The Cosine Rule
C
9
A
A  38 9 
6
5
B
EITHER: Using the Cosine rule for B or C:
e.g.
52  62 - 92
cos B 
 cos B  - 0  333  B  109 5 
2(5)( 6)
Then
C  180 - 109  5 - 38  9  31  6 
OR: Using the Sine rule for C :
5 sin 38  9
sin C sin A
 sin C 

6
c
a
Then
 C  31 6 
B  180 - 31  6 - 38  9  109  5 
The Cosine Rule
SUMMARY
 The Cosine Rule
• In a triangle that isn’t right angled, if we
know 2 sides and the angle formed by the 2
sides, we use
a 2  b 2  c 2 - 2bc cos A
to find the 3rd side.
•
If we know 3 sides, we use
b2  c2 - a 2
cos A 
2bc
to find any angle.