# Cosine Rule maths

All the help you'll need for the cosine rule in this ppt :)

Teacher recommended

- Created by: gogos
- Created on: 23-06-10 11:34

## Slides in this set

### Slide 1

Whiteboardmaths.com

7 2

1 5

© 2004 All rights reserved…read more

### Slide 2

The Cosine Rule

B

Pythagoras' Theorem allows us to

a calculate unknown lengths in

c right-angled triangles using the

relationship a2 = b2 + c2

A b C

It would be very useful to be able to calculate unknown

sides for any value of the angle at A. Consider the square

on the side opposite A when angle A is not a right-angle.

1 2 3

2

a2 a

a2

A A

A Angle A

acute

Angle A

obtuse

a2 = b 2 + c 2 a2 > b 2 + c 2 a2 < b 2 + c 2…read more

### Slide 3

The Cosine Rule

1

The Cosine Rule generalises Pythagoras' Theorem and

takes care of the 3 possible cases for Angle A.

A

Deriving the rule Consider a general triangle ABC. We

require a in terms of b, c and A.

B a2 = b 2 + c 2

BP2 = a2 (b x)2

Also: BP2 = c2 x2 2

c a a2 (b x)2 = c2 x2

a2 (b2 2bx + x2) = c2 x2 A

P a2 b2 + 2bx x2 = c2 x2

A x b b-x C 2 2 2 a2 > b 2 + c 2

a = b + c 2bx*

b

a2 = b2 + c2 2bcCosA 3

Draw BP perpendicular to AC

*Since Cos A = x/c x = cCosA

A

o 2 2 2

When A = 90 , CosA = 0 and reduces to a = b + c 1 Pythagoras

When A > 90o, CosA is negative, a2 > b2 + c2 Pythagoras + a bit

2

a2 < b 2 + c 2

When A < 90o, CosA is positive, a2 > b2 + c2 3 Pythagoras - a bit…read more

### Slide 4

The Cosine Rule

The Cosine rule can be used to find:

1. An unknown side when two sides of the triangle and the included

angle are given.

2. An unknown angle when 3 sides are given.

B

Finding an unknown side.

a2 = b2 + c2 2bcCosA c a

Applying the same method as

A b C

earlier to the other sides

produce similar formulae for b2 = a2 + c2 2acCosB

b and c. namely:

c2 = a2 + b2 2abCosC…read more

### Slide 5

The Cosine Rule a2 = b2 + c2 2bcCosA

To find an unknown side we need 2 sides and the

included angle.

1. Not to 2.

7.7 cm 65o

a 9.6 cm scale 5.4 cm

40o

8 cm m

2 2 2 o m2 = 5.42 + 7.72 2 x 5.4 x 7.7 x Cos 65o

a = 8 + 9.6 2 x 8 x 9.6 x Cos 40

2 2 o m = (5.42 + 7.72 2 x 5.4 x 7.7 x Cos 65o)

a = (8 + 9.6 2 x 8 x 9.6 x Cos 40 )

m = 7.3 cm (1 dp)

a = 6.2 cm (1 dp)

3. 15o 100 m p2 = 852 + 1002 2 x 85 x 100 x Cos 15o

85 m p = (852 + 1002 2 x 85 x 100 x Cos 15o)

p = 28.4 m (1 dp)

p…read more

### Slide 6

The Cosine Rule a2 = b2 + c2 2bcCosA

Application Problem

A fishing boat leaves a harbour (H) and travels due East for 40 miles to a

marker buoy (B). At B the boat turns left onto a bearing of 035o and

sails to a lighthouse (L) 24 miles away. It then returns to harbour.

(a) Make a sketch of the journey

(b) Find the total distance travelled by the boat. (nearest mile)

HL2 = 402 + 242 2 x 40 x 24 x Cos 1250

HL = (402 + 242 2 x 40 x 24 x Cos 1250)

L

= 57 miles

Total distance = 57 + 64 = 121 miles.

24 miles

H

40 miles 125o

B…read more

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