11 x1 t04 06 cosine rule (2013)

Cosine RuleA

B Ca

bc

x

Cosine RuleA

B C

h

a

bc

x

Cosine RuleA

B C

h

a

bc

2 2 2h b x

x

Cosine RuleA

B C

h

a

bc

2 2 2h b x

x

22 2c h a x

Cosine RuleA

B C

h

a

bc

2 2 2h b x

2 2 2 2 22c b x a ax x

x

22 2c h a x

2 2 2b a ax

Cosine RuleA

B C

h

a

bc

2 2 2h b x

2 2 2 2 22c b x a ax x

x

22 2c h a x

2 2 2b a ax

But cos

cos

x Cbx b C

Cosine RuleA

B C

h

a

bc

2 2 2h b x

2 2 2 2 22c b x a ax x

x

22 2c h a x

2 2 2b a ax

But cos

cos

x Cbx b C

2 2 2 2 cosc b a ab C

Cosine RuleA

B C

h

a

bc

2 2 2h b x

2 2 2 2 22c b x a ax x

2 2 2

In any ABC2 cosa b c bc A

x

22 2c h a x

2 2 2b a ax

But cos

cos

x Cbx b C

2 2 2 2 cosc b a ab C

2 2 2 2 cosb a c ac B 2 2 2 2 cosc a b ab C

e.g. iTF

M

t

98 13

3

7

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T 7 2 2 23 7 2 3 7 cos98 13t

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T

8 units (to nearest whole number)t

7 2 2 23 7 2 3 7 cos98 13t

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T

8 units (to nearest whole number)t

( )ii

F

A

S

20

6

7 2 2 23 7 2 3 7 cos98 13t

15

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T

8 units (to nearest whole number)t

( )ii

F

A

S

20

6

2 2 2 2 cosf a s as F 2 2 2

cos2

a s fFas

7 2 2 23 7 2 3 7 cos98 13t

15

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T

8 units (to nearest whole number)t

( )ii

F

A

S

20

6

2 2 2 2 cosf a s as F 2 2 2

cos2

a s fFas

7 2 2 23 7 2 3 7 cos98 13t

15

2 2 26 15 20cos2 6 15

F

e.g. iTF

M

t

98 13

3

2 2 2 2 cost f m fm T

8 units (to nearest whole number)t

( )ii

F

A

S

20

6

2 2 2 2 cosf a s as F 2 2 2

cos2

a s fFas

140 33F

7 2 2 23 7 2 3 7 cos98 13t

15

2 2 26 15 20cos2 6 15

F

iii

B

A

b

257

C3

iii

B

A

b

257

C3

2 2 2 2 cosa b c bc A

2 2 2

2

3 7 2 7 cos 25

9 49 14 cos 25

b b

b b

iii

B

A

b

257

C3

2 2 2 2 cosa b c bc A

2 2 2

2

3 7 2 7 cos 25

9 49 14 cos 25

b b

b b

2 14 cos 25 40 0b b

iii

B

A

b

257

C3

2 2 2 2 cosa b c bc A

2 2 2

2

3 7 2 7 cos 25

9 49 14 cos 25

b b

b b

2 14 cos 25 40 0b b

214cos 25 196cos 25 1602

b

iii

B

A

b

257

C3

2 2 2 2 cosa b c bc A

2 2 2

2

3 7 2 7 cos 25

9 49 14 cos 25

b b

b b

2 14 cos 25 40 0b b

214cos 25 196cos 25 1602

b

5.85 units or 6.85 units (to 2 dp)b b

Exercise 4I; 1a, 2b, 4, 5, 7, 9, 11, 12, 13, 14*

Exercise 4J; evens