The Cosine Rule.

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# The Cosine Rule. - PowerPoint PPT Presentation

C. b. a. A. B. c. a 2 =. b 2. +. c 2. -2bccosA o. The Cosine Rule. C. b. a. h. h. A. B. x. c. D. C. a. c-x. B. D. Proving The Cosine Rule. Consider this triangle:. We are looking for a formula for the length of side “a”. c-x.

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C

b

a

A

B

c

a2 =

b2

+

c2

-2bccosAo

The Cosine Rule.

C

b

a

h

h

A

B

x

c

D

C

a

c-x

B

D

Proving The Cosine Rule.

Consider this triangle:

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

c-x

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

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

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

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

C

b

a

c-x

h

h

A

B

x

c

D

C

a

a2 =

b2

+

c2

-2bccosAo

c-x

B

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

Substitute into the formula:

x =

bcosAo

The Cosine Rule.

10

W

65o

6

6.2

L

89o

13.8

11

8

147o

M

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:

Note the pattern of sides and angle.

L

5m

43o

12m

a2 =

b2

+

c2

-2bccosAo

Using The Cosine Rule.

Example 1.

Find the unknown side in the triangle below:

Identify sides a,b,c and angle Ao

Write down the Cosine Rule.

c =

12

Ao =

43o

a =

L

b =

5

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

17.5 m

137o

12.2 m

M

a2 =

b2

+

c2

-2bccosAo

Example 2.

Find the length of side M.

Identify the sides and angle.

a = M

b = 12.2

C = 17.5

Ao = 137o

Write down Cosine Rule and substitute values.

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

43cm

(1)

78o

31cm

L

6.3cm

(3)

110o

G

(2)

8.7cm

M

5.2m

38o

8m

What Goes In The Box ? 1.

Find the length of the unknown side in the triangles below:

G = 12.4cm

L = 47.5cm

M =5.05m

a2 =

b2

+

c2

-2bccosAo

Finding Angles Using The Cosine Rule.

Consider the Cosine Rule again:

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.

9cm

11cm

xo

16cm

Example 1

Finding An Angle.

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

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-1 0.75 to find Ao

yo

13cm

15cm

26cm

Example 2.

Find the unknown angle in the triangle below:

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 =

- 0.723

Ao =

136.3o

(1)

7m

ao

5m

(3)

14cm

10m

27cm

co

12.7cm

16cm

(2)

8.3cm

7.9cm

What Goes In The Box ? 2

Calculate the unknown angles in the triangles below:

ao =111.8o

bo

bo = 37.3o

co =128.2o