The Cosine Rule

A

B

Ca

b

c

a2 = b

2 + c

2 – 2bc cos A

The Cosine Rule

A

B

Ca

b

c

b2 = a

2 + c

2 – 2cos B

The Cosine Rule

A

B

Ca

b

c

c2 = a

2 + b

2 – 2cos C

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.

6

10

65o

L 89o

13.8

6.2

W

147o

8 11

M

Calculating Sides Using The Cosine

Rule(1).Calculate the length AC

30m

55°

40m

30m

55°

40m

ba

c

b2 = a

2 + c

2 – 2ac cos B

b2 = 30

2+40

2 – 2 30 40 cos 55

b2 = 2500– (2400cos 55)

b2 = 2500– 1376·58

b2 = 2500– 1376·58

b2 = 1123·42

b = 1123·42

b = 33·5 m

Calculating Sides Using The Cosine

Rule(2).

Calculate the length CB

100°13cm5cm

100°13cm5cm

b

a

c

a2 = b

2 + c

2 – 2bc cos A

a2 = 13

2+5

2 – 2 135 cos 100

a2 = 194– (130cos 100)

a2 = 194– (-22·57)

a2 = 216·57

a = 216·57

a = 14·7 cm

a2 = 194– (-22·57)

Rearranging the cosine rule.

a2 = b

2 + c

2 – 2bc cos A

2bc cos A = b2

+c2 – a

2

cosA=b2

+c2–a

2

2bcYou now have a formula for finding an angle if you know all three sides of the triangle.

Calculating Angles Using The Cosine Rule

(1).

Calculate the angle CAB

b

ac

?

cosA=b2

+c2–a

2

2bc

cosA=82

+72–4

2

2 8 7

cosA=82

+72–4

2

2 8 7

cosA=97112

cosA=0·866

A = cos-1 0·866

A = 30°

Calculating Angles Using The Cosine Rule

(2).

Calculate the angle ?

BA

C

cosA=b2

+c2–a

2

2bc

cosA=82

+102–15

2

2 8 10

ba

c

cosA=– 61160

cosA=-0·381

A = cos-1 –0·381

A = 112·4°

cosA=82

+102–15

2

2 8 10