Slides in this set

Slide 1

Preview of page 1

Whiteboardmaths.com
7 2
1 5
© 2004 All rights reserved…read more

Slide 2

Preview of page 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

Preview of page 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

Preview of page 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

Preview of page 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

Preview of page 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

Slide 7

Preview of page 7
Preview of page 7

Slide 8

Preview of page 8
Preview of page 8

Slide 9

Preview of page 9
Preview of page 9

Slide 10

Preview of page 10
Preview of page 10

Comments

daviesg

Report


Lots of illustrated applications of the use of the Sine Rule

EllieDudley

Report

Good job babes

niyi0706

Report

davie more u need a shavey 

Similar Mathematics resources:

See all Mathematics resources »