The Cosine Rule.

A B

C

ab

c

a2 = b2 + c2 -2bccosAo

Proving The Cosine Rule.Consider this triangle:

A B

C

ab

c

We are looking for a formula for the length of side “a”.

Start by drawing an altitude CD of length “h”.

h

D

Let the distance from A to D equal “x”.

x

The distance from D to B must be “c – x”.

c-x

To find the Cosine Rule we are going to concentrate on the triangle “CDB”.

c-x B

C

ah

D

b

A B

C

ab

c

h

D

x c-xc-x B

C

ah

D

Apply Pythagoras to triangle CDB.a2 = h2 + (c - x) 2

Square out the bracket.a2 = h2 + c2 -2cx + x2

What does h2 and x2 make? b2a2 = b2 + c2 -2cx

What does the cosine of Ao equal?

cos Ao = x Make x the subject:

x = bcosAo Substitute into the formula:

a2 = b2 + c2 -2cbcosAo

We now have:

a2 = b2 + c2 -2bccosAo

The Cosine Rule.

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 89o13.8

6.2

W

147o8 11

M

Note the pattern of sides and angle.

Using The Cosine Rule.Example 1.

Find the unknown side in the triangle below:

L5m

12m

43oIdentify sides a,b,c and angle Ao

a = L b = 5 c =12 Ao = 43o

Write down the Cosine Rule.

a2 = b2 + c2 -2bccosAo Substitute values and find a2.

a2 = 52 + 122 - 2 x 5 x 12 cos 43o

a2 = 25 + 144 - (120 x 0.731 )

a2 = 81.28 Square root to find “a”.a = 9.02m

Example 2.

137o17.5 m

12.2 m

M

Find the length of side M.

Identify the sides and angle.

a = M b = 12.2 C = 17.5 Ao = 137oWrite down Cosine Rule and substitute values.

a2 = b2 + c2 -2bccosAo

a2 = 12.22 + 17.52 – ( 2 x 12.2 x 17.5 x cos 137o )

a2 = 148.84 + 306.25 – ( 427 x – 0.731 ) Notice the two negative signs.a2 = 455.09 + 312.137

a2 = 767.227

a = 27.7m

What Goes In The Box ? 1.Find the length of the unknown side in the triangles below:

(1)78o

43cm

31cmL

(2)

8m

5.2m

38o

M

(3) 110o

6.3cm

8.7cm

G

L = 47.5cm

M =5.05m

G = 12.4cm

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

Take b2 and c2 across.-2bc cos Ao = a2 – b2 – c2

Divide by – 2 bc.bc

cbaAo

2cos

222

Divide top and bottom by -1

bc

acbAo

2cos

222

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

Finding An Angle.

Use the formula for Cos Ao to calculate the unknown angle xo below:

xo

16cm

9cm 11cm

Write down the formula for cos Ao

bc

acbAo

2cos

222

Identify Ao and a , b and c.

Ao = xo

a = 11 b = 9 c = 16

Substitute values into the formula.

1692

11169cos

222

oA

Calculate cos Ao .

Cos Ao = 0.75

Use cos-1 0.75 to find Ao

Ao = 41.4o

Example 1

Example 2.

Find the unknown angle in the triangle below:

26cm

15cm 13cmyo Write down the formula.

bc

acbAo

2cos

222

Identify the sides and angle.

Ao = yo a = 26 b = 15 c = 13

Substitute into the formula.

13152

261315cos

222

oA Find the value of cosAo

cosAo = - 0.723 The negative tells you the angle is obtuse.

Ao = 136.3o

What Goes In The Box ? 2Calculate the unknown angles in the triangles below:

(1)

10m

7m5m ao

bo

(2)12.7cm

7.9cm 8.3cm

(3)

co27cm

14cm

16cm

ao =111.8o

bo = 37.3o

co =128.2o