NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y...

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NELSON SENIOR MATHS SPECIALIST 11 FULLY WORKED SOLUTIONS CHAPTER 9: Trigonometric identities Exercise 9.01: Reciprocal trigonometric functions Concepts and techniques 1 cot (θ) is the reciprocal of tan (θ) so cot 3 π = 1 3 A 2 cosec (θ) is the reciprocal of sin (θ) so cosec 4 π = 2 D 3 sec (θ) is the reciprocal of cos (θ) so sec 6 π = 2 3 D 4 ( ) cosec 90° = 1 D 5 ( ) cot 45° = 1 C all cos tan sin all sec cot cosec Positve in quadrants.....

Transcript of NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y...

Page 1: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

NELSON SENIOR MATHS SPECIALIST 11 FULLY WORKED SOLUTIONS

CHAPTER 9: Trigonometric identities Exercise 9.01: Reciprocal trigonometric functions

Concepts and techniques

1 cot (θ) is the reciprocal of tan (θ) so cot3π

= 13

A

2 cosec (θ) is the reciprocal of sin (θ) so cosec4π

= 2 D

3 sec (θ) is the reciprocal of cos (θ) so sec6π

= 23

D

4 ( )cosec 90° = 1 D

5 ( )cot 45° = 1 C

all

costan

sin all

seccot

cosec

Positve in quadrants.....

Page 2: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

6 a cot( ) cot4 4π π π − = −

= –1

b sec sec 24 4π π π − = − = −

c 2cosec sin3 3 3π π π + = − = −

d sec 2 sec 23 3π π π − = − =

e cot cot 36 6π π π + = =

f sec sec 24 4π π π + = − = −

g sec 2 sec 24 4π π π + = =

h cot 2 cot 36 6π π π + = =

7 a ( )sec(2 ) secπ + θ = θ

b ( )sec( ) secπ − θ = − θ

c ( ) ( )cosec(2 ) cos ec cos ecπ − θ = −θ = − θ

d ( )cot( ) cotπ + θ = θ

e ( )cosec( ) cosecπ + θ = − θ

f ( ) ( )cot(2 ) cot cotπ − θ = −θ = − θ

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Page 3: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

8 For 0 2x π≤ ≤ sketch the graph of ( )3cosec 1y x= −

9 For 0 2x≤ ≤ π sketch the graph of sec 12

y x π = − − +

10 For 0 2x≤ ≤ π sketch the graph of ( )4cot 2 1y x= +

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Page 4: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

11 For x−π ≤ ≤ π sketch the graph of 2sec 22

y x π = − +

Reasoning and communication

12 sec 24π =

and cosec 2

4π =

Equal values.

NB 4π +

4 2π π=

NB cos4π =

sin

13 sec3π

= 2 and cosec 26π =

NB. 3π +

6 2π π=

NB. cos3π =

sin

NB. cos (θ) = sin 2π − θ

so sec (θ) = cosec

2π − θ

© Cengage Learning Australia 2014 ISBN 9780170251501 4

Page 5: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

14 a

π 2sec6 3

=

b πcot 14

=

c π 2cosec3 3

− = −

d 2π π 2cosec cos ec3 3 3

= =

e sec (0) = 1

f cot (0) is not defined as tan (0) = 0

g 4π π 1cot cot3 3 3

− = − = −

h 7π πsec sec 24 4

− = =

i 3πcosec 12

= −

j πsec3

= 2

15 Use the graph of y = tan (x) to sketch y = cot (x) from − π2

to π2

.

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Page 6: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from − π2

to π2

.

17 Use the graph of y = sin (4x) to sketch the graph of y = cosec (4x) from 0 to π.

18 Use the graph of y = tan (2x) to sketch the graph of y = cot (2x) from −π to π.

19 Use the graph of πcos4

y x = −

to sketch the graph of πsec4

y x = −

from −π to π.

© Cengage Learning Australia 2014 ISBN 9780170251501 6

Page 7: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

20 Use the graph of πsin6

y x = +

to sketch the graph of πcosec6

y x = +

from

0 to 2π.

21 Use the graph of πtan4

y x = +

to sketch the graph of πcot4

y x = +

from

− π2

to π2

.

22 Sketch the graph of πcosec4

y x = +

from 0 to 2π.

© Cengage Learning Australia 2014 ISBN 9780170251501 7

Page 8: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

23 Sketch the graph of πsec 34

y x = +

from 0 to π.

24 Sketch the graph of πcot 24

y x = −

from 0 to π.

25 Sketch the graph of πsec 23

y x = −

from 0 to π.

© Cengage Learning Australia 2014 ISBN 9780170251501 8

Page 9: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.02: The Pythagorean identities

Concepts and techniques

1 Prove that sin2 (A) + cos2 (A) 1≡ .

Using the equation 2 2 2c a b= + we get the relationship 2 2 2r x y= +

We know that ( )cos xr

θ = and ( )sin yr

θ =

( ) ( )cos cosx x rr

θ = ⇒ = θ and ( ) ( )sin siny y rr

θ = ⇒ = θ

giving ( ) ( )2 2 2 2 2cos sinr r r= θ + θ

So for any r , ( ) ( )2 21 cos sin= θ + θ

2 Prove that ( ) ( )( )

sintan

cosA

AA

≡ .

Consider the diagram showing angle θ, general point P(x, y) and radius of circle, r.

Using ( )tan yx

θ = then we have ( ) ( )( )

sintan

cosθ

θ ≡θ

i.e. ( ) ( )( )

sintan

cosA

AA

© Cengage Learning Australia 2014 ISBN 9780170251501 9

Page 10: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

3 Prove that 2 21 cot ( ) cosec ( )A A+ ≡ .

( ) ( )

( )( )

( )( )

( ) ( )( )

( )( )

22

2

2

2

2

2 2

2

2

2

11 cot 1tan

11sin

cos

cos1

sin

sin +cossin

1sin

= cosec A

AA

AA

AA

A AA

A

+ ≡ +

= +

= +

=

=

4 Prove that ( ) ( )( )

coscot

sinA

AA

.

( ) ( )

( )( )

( )( )

1cottan

1sin

cos

cossin

AA

AA

AA

=

=

=

5 Show that

2sin2 3tan

23 cos3

π π = π

.

Given ( ) ( )( )

sintan

cosA

AA

≡ then if A = 23π then

2sin2 3tan

23 cos3

π π = π

© Cengage Learning Australia 2014 ISBN 9780170251501 10

Page 11: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Reasoning and communication

6 Prove that [1 − sin (θ)][1 + sin (θ)] = cos2 (θ).

Given sin2 (A) + cos2 (A) 1≡ then

cos2 (θ) = 1 – sin2 (θ)

= [1 − sin (θ)][1 + sin (θ)]

7 Prove that sin4 (θ) − cos4 (θ) = 1 − 2 cos2 (θ).

sin4 (θ) − cos4 (θ) = [sin2 (θ) +cos2 (θ)][sin2 (θ) – cos2 (θ)]

= 1 × {[1 – cos2 (θ)] – cos2 (θ)}

= 1 − 2 cos2 (θ).

8 Prove that ( ) ( ) ( )( )

2 1 coscot cosec .

1 cosx

x xx

++ = −

( ) ( ) ( )( ) ( )

( )( )

( )( )( ) ( )( ) ( )( )( )

22

2

2

2

cos 1cot cosecsin sin

cos 1sin

cos 11 cos

1 cos 1 cos

1 cos 1 cos

1 cos.

1 cos

xx x

x x

xx

xx

x xx x

xx

+ = +

+=

+ =−

+ + =− + +

=−

9 Prove that ( ) ( ) ( )( )

1 sinsec tan .

cosy

y yy

++ =

( ) ( ) ( )

( )( )

( )( )

sin1sec tancos cos

1 sincos

yy y

y y

yy

+ = +

+=

© Cengage Learning Australia 2014 ISBN 9780170251501 11

Page 12: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

10 Prove that ( ) ( ) ( )2 1 12 cosec .

1 cos 1 cosa

a a= +

− +

( ) ( )( ) ( )( ) ( )

( ) ( )

( )

( )( )

2

2

2

1 cos 1 cos1 11 cos 1 cos 1 cos 1 cos

21 cos 1 cos

21 cos

2sin

2cos ec

a aa a a a

a a

a

a

a

+ + −+ =

− + − +

=− +

=−

=

=

11 Prove that ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

sin cos βsin β 1 tan β cos β 1 cot β .

cos β sin ββ +

= + + +

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( )

( ) ( ) ( )( ) ( ) ( ) ( )

( )

( ) ( ) ( )( )

( )( )

( ) ( ) ( ) ( )( ) ( )

2

sin β cos βsin β 1 tan β cos β 1 cot β sin β 1 cos β 1

cos sin

cos sin sin cossin β cos

cos sin

sin cossin β cos

cos sin

sin cos 2βsin β cos

cos sin

si

+ + + = + + + β β

β + β β + β= + β β β

β β= + β + β β

β += + β β β

=( ) ( )( ) ( )

n cos βcos β sin β

β +

© Cengage Learning Australia 2014 ISBN 9780170251501 12

Page 13: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

12 Prove that ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )sin θ cos θ cos θ cot θ sin θ tan θ

1 sin θ cos θ .cos θ sin θ

− + =−

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

3 3

3 3

2

sin θ cos θ cos θ cot θ sin θ tan θcos θ sin θ

sin θ cos θ cos θ sin θcos θ sin θ

cos θ sin θ sin θ cos θ

sin θ cos θ cos θ sin θcos θ sin θ sin θ cos θ

cos θ sin θcos θ sin θ

cos θ sin θ cos θ +cos θ sin θ sin

− −

= − − −

= − −

=−

− + ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

2

2 2

θ

cos θ sin θ

cos θ +cos θ sin θ sin θ

1 sin θ cos θ .

= +

= +

© Cengage Learning Australia 2014 ISBN 9780170251501 13

Page 14: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.03: Angle sum and difference identities

Concepts and techniques

1 a sin (2x + y) = sin (2x) cos (y) + sin (y) cos (2x)

b cos (4x − 3y) = cos (4x) cos (3y) + sin (4x) sin (3y)

c sin (3p − 5q) = sin (3p) cos (5y) – cos (3p) sin (5y)

d tan (2e − f) = ( ) ( )( ) ( )

tan 2 tan1 tan 2 tan

e fe f−

+

e cos (3α + 2β) = ( ) ( ) ( ) ( )cos 3 cos 2 sin 3 sin 2α β − α β

f sin (4m + n) = ( ) ( ) ( ) ( )sin 4 cos cos 4 sinm n m n+

g tan (w + 3x) = ( ) ( )( ) ( )

tan tan 31 tan tan 3

w xw x+

h sin (2D − 5E) = ( ) ( ) ( ) ( )sin 2 cos 5 cos 2 sin 5D E D E−

i cos (P − 4Q) = cos (P) cos (4Q )+ sin (P) sin (4Q)

2 a sin (π − x) = sin (π) cos (x) − sin (x) cos (π) = 0 – sin (x) × (–1) = sin (x)

b cos (π + x) = cos (π) cos (x) − sin (x) sin (π) = –1 × cos (x) – 0 = – cos (x)

c ( )πtan cot2

x x − =

d

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )π π π2 2 2

π 1 1 1 1cosec sec2 sin sin cos cos sin 0 cos 1 cos

x xx x x x x

+ = = = = = + + + ×

© Cengage Learning Australia 2014 ISBN 9780170251501 14

Page 15: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

e

3π πcot cot2 2

x x − = π + −

Consider tan π2

x π + −

( ) ( )( ) ( )( )

( )( )( )( )( )( )

π2π2

π2

π2

π2π2

tan tanπtan2 1 tan tan

0 tan1 0

tan

sincos

cossin

cot

xx

x

x

x

xx

xx

x

π + − π + − = − π −

+ −=

−= −

−=

=

=

( )

( )

3π 1cot2 cot

3πcot tan2

xx

x x

− =

− =

f πsec2

x +

( ) ( )

( )( )

( )

( )

π2

π π πcos co

Cons

s

ider c

sin sin2 2 2

0 sin 1

sin

π 1sec2 sin

πsec

os

c

cos ec2

os

x

x x x

x

x

xx

x x

+

+ = −

= − ×

= −

∴ + = −

∴ + = −

g sin (2π − x) = sin(–x) = –sin (x)

© Cengage Learning Australia 2014 ISBN 9780170251501 15

Page 16: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

h ( ) ( )3π 3π 3πcos cos cos sin sin2 2 2

x x x − = +

( )( )

0 ( 1) sin

sin

x

x

= + − ×

= −

i tan (x − π) ( ) ( )( ) ( )

( ) ( )tan tan tan 0

tan1 tan tan 1 0

x xx

x− π −

= = =+ π −

3 Using ( ) ( ) ( ) ( )cos( ) cos cos sin sinx y x y x y− = + and putting 2

x π=

( ) ( )

( ) ( )

( )

cos cos cos sin sin2 2 2

0 cos 1 sin

cos sin2

x x x

x x

x x

π π π − = +

= × + ×

π − =

Using ( ) ( ) ( ) ( )sin( ) sin cos cos sinx y x y x y− = − and putting 2

x π=

( ) ( )

( ) ( )

( )

sin sin cos cos sin2 2 2

1 cos 0 sin

sin cos2

x x x

x x

x x

π π π − = −

= × − ×

π − =

4

( )( )

( )( )

( )( ) ( )

π π2 2π π2 2

sin sin costan cot

2 cos cos sinx x x

x xx x x

+ −π + = = = = − + − − −

© Cengage Learning Australia 2014 ISBN 9780170251501 16

Page 17: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Reasoning and communication

5 Prove that sin (x + y) = sin (x) cos (y) + cos (x) sin (y). Given ( ) ( ) ( ) ( )cos( ) cos cos sin sinx y x y x y− = + found using cosine laws and distance formula

Putting y = 2

yπ− we have ( ) ( ) ( ) ( )cos( ) cos cos sin sinx y x y x y− = +

becomes

cos 2

x y π − − = cos (x) cos

2yπ −

+ sin (x) sin

2yπ −

i.e. cos2

x y π − − − = cos (x) sin (y) + sin (x) cos (y) but cos(–θ) = cos (θ)

cos2

x yπ − −

= cos (x) sin (y) + sin (x) cos (y)

∴ sin (x + y) = sin (x) cos (y) + cos (x) sin (y).

6 Prove that ( ) ( ) ( )( ) ( )

tan tantan .

1 tan tanx y

x yx y−

− =+

( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( )( )

( )( )

( )( )

( )( )

( ) ( )( ) ( )

sin( )tancos( )

1sin cos sin cos cos cos

1cos cos sin sincos cos

sin cos sin coscos cos

cos cos sin sincos cos

sin sincos cos

sin sin1

cos cos

tan tan1 tan tan

x yx yx y

x y y x x yx y x y

x y

x y y xx y

x y x yx y

x yx y

x yx y

x yx y

−− =

−= ×

+

=−

−=

+

−=

+

© Cengage Learning Australia 2014 ISBN 9780170251501 17

Page 18: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

7 Prove that ( ) ( ) ( )( ) ( )

cot cot 1cot .

cot cotx y

x yy x

−+ =

+

Consider tan(x+y)

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )

( ) ( )( )

( ) ( )( )

( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

tan tan1 tan tan

11 tan tan tan tan

cot1tan tan

tan tan

1 tan tantan tan

tan tantan tan

1 1tan tan

tan tantan tan tan tan

1 1tan tan

1 1tan tan

cot cot 1

ta

t t

n

co co

x yx y

x y x yx y

x y

x yx y

x yx y

x yx y

x y x y

x y

x y

y

x

y x

x

y

x y

+=

−∴ = ×

+

=+

−=

+

+

+

=+

+

=

8 Prove that ( ) ( ) ( ) ( )( )

sec sectan tan .

cosecx y

x yx y

+ =+

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )

( )( )

( )( )

( ) ( )

sec sec 1sin( )cosec cos cos

sin cos sin coscos cos

sin sincos cos

tan tan

x yx y

x y x y

x y y xx y

x yx y

x y

= + ×+

+=

= +

= +

© Cengage Learning Australia 2014 ISBN 9780170251501 18

Page 19: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

9 Prove that sin (x + y) + cos (x − y) = (sin (x) + cos (x)(sin (y) + cos (y)).

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

(

sin cos sin cos cos cos sin sin

sin

s

cos sin cos sin cos

cos sin sin cos

si

in cos

n cos cos s

)

in

x y x

x y y x x y x y

x y y x y y

y y x x

x x y

y

y

= + + +

= + + + = − + = + −

+ +

10 Prove that sin (x + y) sin (x − y) = sin2 x − sin2 y.

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2 2

sin cos cos sin sin cos cos sin

sin cos sin cos cos sin cos sin sin cos cos s

sin si

in

sin cos cos sin

sin 1 sin 1 sin sin

sin sin sin sin si s n

n

n i

x y x y x y x y

x y x y x y x y x y x y

x y x y

x y x y

x x y y

x x y

x

y

= + − = − + −

= −

= − − − +

= −

+

− ( )( ) ( )

2

2 2sin sin

y

x y= −

11 Prove that ( ) ( )( ) ( )

( )( )

cot cot sin.

1 cot cot cosx y x y

x y x y− −

=− +

( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

1sin sin cos cos sin sin sin

1cos cos cos sin sinsin sin

sin cos cos sinsin sin

cos cos sin sinsin sin

cot cot 1cot cot 1 1

cot cot1 cot cot

x y x y x y x yx y x y x y

x y

x y x yx y

x y x yx y

y xx y

x yx y

− −= ×

+ −

=−

− −= ×

− −

−=

© Cengage Learning Australia 2014 ISBN 9780170251501 19

Page 20: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

12 Prove that tan (x − y) + tan (y − z) + tan (z − x) = tan (x − y) tan (y − z) tan (z − x).

( ) ( ) ( )( )( )

( )( )

( )( )

( ) ( ) ( ){ } ( ) ( ) ( ){ } ( ) ( ) ( )( ) ( ) ( )

( ){ } ( ) ( ) ( ) ( ){ } ( ) ( ) ( )

sin sincos cos cos

cos cos sin cos cos sin cos coscos cos cos

cos c

tan tan tan

s

os sin cos sin co

in

sin

sin s cosco

x y y z z x

x y y z z xx y y z z x

x y y z z x y z x y z x z x x y y zx y y z z x

z x x y y z y z x y z x x y y z

− + − + −

− − −+ +

− − −

− − − − − − − − −

− − −

− − −

=

+ +=

+ +=

− − − − −

( ) ( ) ( )( ){ } ( ) ( ) ( ) ( ){ } ( ) ( ) ( )

( ) ( ) ( )( ){ }

s cos cos

cos cos cos sin sin cos coscos cos cos

co

sin

sis n

x y y z z x

z x x y y z x y y z z x x y y zx y y z z x

z x x y

− − −

− − − − − − − −+ +=

=

− − −

− − y+( ){ } ( ) ( ) ( )( ) ( ) ( )

( ){ } ( ){ } ( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )( )

( ) ( ) ( )

sin cos cos

cos cos cos

cos sin cos coscos cos cos

and given that sin( ) sin

sin cos sin cos coscos cos cos

sin cos

s

cos coscos

in

z z x x y y z

x y y z z x

z x x z z x x y y zx y y z z x

z x z x z x x y y zx y y z z x

z x z x x y y zz x

− − − −

− − −

− − − − −

− − −

− − −

+

+=

− α = − α

+=

− +=

− −− − −

− − − −− ( ) ( )

( )( ) ( )( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( )( ) ( )

cos cos

cos cos costan

cos cos

cos cos costan

cos cos

costan

cos sin sin

cos

x y y z

z y y x x y y zz x

x y y z

z y y x z y y x x y y zz x

x y y z

z y y xz x

− +

= −

− −

− + − − −−

− −

− − + − +=

−=

− −−

− −

− −−

( ) ( ) ( ) ( )sin sin cos cosz y y x x y y z++ − − − −

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

cos cos

tancos cos

tancos cos

tan tan tan given sin = -sin and sin = -s

sin sin

sin sin

tan tan

in

tan tantan

x y y z

z y y xz x

x y y z

y x z yz x

x y y z

z x y x y z y x x y z y y z

x y y z z x x y y x

− −

− −−

− −

− −−

− −

− − − − − − −

− + −

=

=

=

−∴ =+ − − ( )tan z x−

© Cengage Learning Australia 2014 ISBN 9780170251501 20

Page 21: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.04: Double angle formulas

Concepts and techniques

1 Show that ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

2 2

2 2

2

cos 2 cos sin from the formula from cos ( ) = cos cos sin sin

= 1 sin sin

1 2sin

A A A A A A A A A

A A

A

= − + −

− − = −

2 Show that ( ) ( )2 22cos 1 1 2sinA A− = −

( ) ( )( )( )

2 2

2

2

2cos 1 2 1 sin 1

2 2sin 1

1 2sin

A A

A

A

− = − − = − −

= −

3 Show that ( ) ( ) ( ) ( )2 2 2 2cos sin 1 sin siny y A A − = − −

( )21 2sin A= −

4 Show that ( ) ( ) ( )sin 8 2sin 4 cos 4A A A= .

( )

( ) ( ) ( ) ( )( ) ( )

sin 8 sin(4 4 )

sin 4 cos 4 sin 4 cos 4

2sin 4 cos 4

A A A

A A A A

A A

= +

= +

=

5 Show that ( ) 2 5cos 5 1 2sin2AA = −

( )

2 2

2 2

2

5 52 2

5 5 5 5cos cos sin sin2 2 2 25 5cos

cos

sin2 2

5 51 sin sin2 2

51 2s

5 cos

2

in

A A

A A A A

A A

A A

A

A +

= − = −

= − −

=

= −

© Cengage Learning Australia 2014 ISBN 9780170251501 21

Page 22: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

6 a sin (4x) = 2sin (2x) cos (2x)

b cos (6A) = 2cos2(3A) – 1

c ( ) ( )( )2

2 tan 2tan 4

1 tan 2y

yy

=−

d ( ) 2 2cos cos sin2 2y yy = −

e 2

2 tan(2 )tan(4 )1 tan (2 )

xxx

=−

7 a sin (4x) = 2 sin (2x) cos (2x)

= 2(2 sin (x)cos (x)[cos2 (x) – sin2 (x)]

= 4 sin (x) cos3(x) – 4 sin3(x) cos (x)

b cos (6A) = cos2(3A)– sin2(3A)

= (4cos3(A)–3cos (A))2 – (3sin (A) – 4cos3(A))2

= (16cos6(A) – 24 cos4(A) + 9cos2(A)) – (9sin2(A) – 24 sin4(A) +16sin6(A))

= 16cos6(A) – 16sin6(A)+ 9 cos2(A) – 9sin2(A) – 24(cos4(A)– sin4(A))

= 16(cos2(A) – sin2(A))(cos4(A)+ sin2(A)cos2(A) + sin4(A))+ 9(cos2(A)

– sin2(A)) – 24(cos2(A)– sin2(A)) (cos2(A)+ sin2(A))

= (cos2(A) – sin2(A))[16(cos4(A)+ sin2(A)cos2(A) + sin4(A)) + 9 – 24]

= (cos2(A) – sin2(A))[16(cos4(A)+ sin2(Acos2(A) + sin4(A)) – 15]

= (cos2(A) – sin2(A))[16(cos4(A)+2sin2(A)cos2(A) + sin4(A)– sin2(A)cos2(A)) – 15]

= (cos2(A )– sin2(A))[16(1– sin2(A)cos2(A) )– 15]

= (cos2(A) – sin2(A))[16 – 16sin2(A)cos2(A) – 15]

= (cos2(A) – sin2(A))[1 – 16sin2(A)cos2(A)]

= cos2(A) – sin2(A) – 16sin2(A)cos4(A) + 16sin4(Acos2(A)

NB. There are alternative answers. Answers may vary.

Sometimes answers are given in terms of cos (x) or sin (x) only.

© Cengage Learning Australia 2014 ISBN 9780170251501 22

Page 23: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

c

( ) ( )( )( )( )( )( )

( )( )( )( )( )( )

( ) ( )( )

( )( )

( )( )

( ) ( ) ( )( )

2

2

2

2

2

2

22

2

22 2

22

2

2 2

2 2 2

22

2 tan 2tan 4

1 tan 2

2[2 tan1 tan

2 tan1

1 tan

4 tan1 tan

4 tan1

1 tan

4 tan1 tan

1 tan 4 tan

1 tan

4 tan 1 tan1 tan 1 tan

1 2 tan tan 4 4 tan

1 tan

4 t

yy

y

yy

yy

yyy

y

yy

y y

y

y yy y

y y y

y

=−

−=

−=

− −

−= − −

− −

×− −

=− + −

=( ) ( )

( ) ( )

2

2 2

an 1 tan

1 6 tan tan 4

y y

y y

− − +

© Cengage Learning Australia 2014 ISBN 9780170251501 23

Page 24: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

8 Prove that ( ) ( ) ( ) ( ) ( ) ( )1 1 2cos sec 2

cos sin cos sin+ = θ θ

θ + θ θ − θ

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )( ) ( )( )

( )( ) ( )

2 2

1 1cos sin cos sin

cos sin cos sin

cos sin cos sin

2coscos sin cos sin

2coscos sin

2coscos 2

2cos sec 2

+θ + θ θ − θ

θ − θ + θ + θ =θ + θ θ − θ

θ=

θ + θ θ − θ θ

= θ − θ

θ=

θ

= θ θ

9 Prove that ( ) ( ) ( ) ( )( )

sec 4 sectan 4 tan .

cosec 3x x

x xx

− =

( ) ( ) ( )( )

( )( )

( ) ( ) ( ) ( )( ) ( )

( ) ( ) [ ]

( ) ( ) ( )

sin 4 sintan 4 tan

cos 4 cos

sin 4 cos cos 4 sincos 4 cos

1 sin(4 )cos 4 cos

sec 4 sec sin 3

x xx x

x x

x x x xx x

x xx x

x x x

− = −

−=

= −

=

10 Show that sin (4x) = 4sin (x) cos (x) – 8sin3(x)cos (x)

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

2

3

sin 4 2sin 2 cos 2

2 2sin cos 1 2sin

4sin cos 8sin cos

x x x

x x x

x x x x

=

= − = −

© Cengage Learning Australia 2014 ISBN 9780170251501 24

Page 25: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

11 Show that ( ) ( )( )

2cot 1cot 2 .

2cotx

xx−

=

( ) ( )( )

( ) ( ) ( )

( ) ( )( )( )

( )( )

2

2

2

1 tan1 1 1cot 2 tantan 2 2 tan 2 tan

cot 11 1 1cot2 cot 2 cot

cot 1

2cot

xx x

x x x

xx

x x

xx

−= = = −

− = − =

− =

12 Prove that cot (θ) − tan (θ) = 2 cot (2θ).

2 cot (2θ) = 2 × ( )( )

2cot 12cot

θ −θ

= ( )( )

( )( ) ( ) ( ) ( )

2 2cot 1 cot 1 cot tancot cot cot

θ − θ= − = θ − θ

θ θ θ

13 Prove that ( ) ( ) ( )1 1tan 2 .

1 tan 1 tanx

x x= −

− +

( ) ( )( ) ( )( ) ( )( )( )

( )

2

1 tan 1 tan1 11 tan 1 tan 1 tan 1 tan

2 tan1 tan

tan 2

x xx x x x

xx

x

+ − +− =

− + − +

=−

=

14 Prove that tan (x) = cot (x) − 2 cot (2x).

( ) ( ) ( )( )

( )( )

( )( )

2

2

1 tan1cot 2 cot 2 2tan 2 tan

1 1 tantan

tan

xx x

x x

xx

x

−− = −

− +

=

=

© Cengage Learning Australia 2014 ISBN 9780170251501 25

Page 26: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.05: Finding and using exact values

Concepts and techniques

1 a tan (15°) = tan (45° − 30°)

( ) ( )( ) ( )

( )( )

13

13

113

1 13

2 1 4 23 33 3

1 23 3

1tan 45 tan 301 tan 45 tan 45 1 1

1

1

4 2 3 33 23 3

4 2 3 3 4 2 33 3 2 2 2

2 3

−° − °= = ×

+ ° ° + ×

− + −= =

= − × ×

= − × = −

= −

b ( )cos 75 cos (45 30 )° = ° + °

( ) ( ) ( ) ( )

( )

cos 45 cos 30 sin 45 sin 30

1 3 1 12 22 2

3 1 22 2 2

2 3 1

46 2

4

= ° ° − ° °

= × − ×

−= ×

−=

−=

© Cengage Learning Australia 2014 ISBN 9780170251501 26

Page 27: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Conversion between answers in textbook.

2 2

2

4(2 3)2 32 4

8 4 34

6 2 4 3 24

( 6) 2 6 2 ( 2)4

( 6 2)4

6 24

−−=

−=

− × +=

− × +=

−=

−=

c 7 4 3tan tan12 12 12π π π = +

( )

tan tan3 4

1 tan tan3 4

3 1 1 31 3 1 31 2 3 3

1 32 2 3

22 3

π π + =

π π −

+ += ×

− +

+ +=

+=

−= − −

© Cengage Learning Australia 2014 ISBN 9780170251501 27

Page 28: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

d

5π π πsin sin12 6 4 = +

( )

π πsin cos sin cos6 4 4 6

1 1 1 32 22 21 3 22 2 2

2 1 3

4

π π = +

= × + ×

+= ×

+=

e

11π 3π 8π π 2πcot cot cot12 12 12 4 3

= + = +

( )( )

π 2πConsider tan4 3

2tan tanπ 2π 4 3tan

24 3 1 tan tan4 3

1 31 31 1 3 1 3

21 2 3 3

12 3

π 2πcot 3 24 3

+

π π + + = π π −

+−= ×

+ × +

−=

+ +−

=+

∴ + = − −

© Cengage Learning Australia 2014 ISBN 9780170251501 28

Page 29: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

f sin (105°) = sin (60° + 45°)

= sin (60°) cos (45°) + sin (45°) cos (60°)

( )

3 1 1 12 22 23 1 2

2 2 2

2 3 1

4

= × + ×

+= ×

+=

g cos (195°) = cos (150° + 45°)

= cos (150°) cos (45°) – sin (45°) sin (150°)

( )

3 1 1 12 22 2

3 1 22 2 2

2 3 1

4

= − × − ×

+= − ×

+= −

h

7πsin12

= sin 4π 3π π πsin

12 12 3 4 + = +

= sin π3

cos π4

+ sin π4

cos π3

( )

3 1 1 12 22 23 1 2

2 2 2

2 3 1

4

= × + ×

+= ×

+=

© Cengage Learning Australia 2014 ISBN 9780170251501 29

Page 30: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

i sin (195°) = sin (150° + 45°)

= sin (150°) cos (45°) + sin (45°) cos (150°)

( )

1 1 1 32 22 21 3 22 2 2

2 1 3

4

−= × + ×

−= ×

−=

© Cengage Learning Australia 2014 ISBN 9780170251501 30

Page 31: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

2 ( )tan 75° = tan (x + y) where x = 45° and y = 30°

( ) ( )( ) ( )( ) ( )( ) ( )

13

13

13

13

3 13

3 13

tan tantan( )

1 tan tan

tan 45 tan 30tan(45 30 )

1 tan 45 tan 301

1 1

1

1

3 1 33 3 1

3 13 13 1 3 13 1 3 1

3 2 3 13 1

4 2 32

2 3

x yx y

x y

+

++ =

° + °° + ° =

− ° °

+=

− ×

+=

=

+= ×

+=

+ += ×

− +

+ +=

−+

=

= +

© Cengage Learning Australia 2014 ISBN 9780170251501 31

Page 32: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

3 a sin ?12π =

1

12 2 6π π= ×

Use cos (2x) = 1 – 2 sin2(x)

( ) ( )

( )

2 2

2

2 3 2 2 32 4

8 4 34

6 2 12 24

6 2 6 2 2

4

6 2

46 2

4

− −=

−=

− +=

− +=

−=

−=

b 1cos ?8 8 2 4π π π = = ×

2

2

2

cos 2cos 14 8

2cos cos 18 4

1 1cos 18 2 2

1 2 2cos8 2 2 2

but 0 cos8

2 2so cos8 4

2 22

π π = −

π π = + π = +

π + = ± ×

π ≤

π + =

+=

© Cengage Learning Australia 2014 ISBN 9780170251501 32

Page 33: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

c 5 5 3tan ?12 12 4 3π π π π = = −

( ) ( )( ) ( )

tan tanUse tan( )

1 tan tan

3tan tan5 4 3tan

312 1 tan tan4 3

1 31 ( 1) 3

1 3 1 31 3 1 31 2 3 3

1 34 2 3

22 3

x yx y

x y−

− =+

π π − π ∴ = π π +

− −=

+ − ×

− − += ×

− +

− − −=

−− −

=−

= +

d 1sin ?8 8 2 4π π π = = ×

2

2

2

cos 1 2sin4 8

2sin 1 cos8 4

1 1sin 18 2 2

2 1 2sin8 2 2 2

but sin 08

2 2so sin8 4

2 22

π π = −

π π = − π = −

π − = ± ×

π >

π − =

−=

© Cengage Learning Australia 2014 ISBN 9780170251501 33

Page 34: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

e 5cos cos12 6 4π π π = +

( )

cos cos sin sin6 4 6 4

3 1 1 12 22 23 1 2

2 2 2

2 3 1

4

π π π π = −

= × − ×

−= ×

−=

f 1tan ?8 8 2 4π π π = = ×

Use ( ) ( )( )2

2 tan1 a

t n 2t

an

AA

A =−

and tan4π

= 1

2

2

2

2

2 tan81

1 tan8

1 tan 2 tan8 8

tan 2 tan 1 08 8

2 2 4( 1)1tan

8 2

2 82

2( 1 2)tan8 2

tan 08

tan 2 18

π =

π −

π π − =

π π + − =

− ± − −π =

− ±=

π − ± = π >

π = −

© Cengage Learning Australia 2014 ISBN 9780170251501 34

Page 35: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.06: Products to sums

Concepts and techniques

1 a ( ) ( )cos 10 cos 50° ° = ? given ( )sin 40 0.6428° ≈ and ( )cos 40 0.7660° ≈

( ) ( ) [ ]( ) ( ) [ ]

( )

12

12

12

12

Use cos cos cos( ) cos( )

so cos 10 cos 50 cos( 40 ) cos(60 )

0.7660 0.51.2660

0.6330

A B A B A B= × − + +

° ° = × − ° + °

= × +

= ×

=

b ( ) ( )sin 10 sin 50° ° = ? given ( )sin 40 0.6428° ≈ and ( )cos 40 0.7660° ≈

( ) ( ) [ ]( ) ( ) [ ]

( )

12

12

12

12

Use sin sin cos( ) cos( )

so sin 10 sin 50 cos( 40 ) cos(60 )

0.7660 0.50.2660

0.1330

A B A B A B= × − − +

° ° = × − ° − °

= × −

= ×

=

c ( ) ( )sin 10 cos 50° ° = ? given ( )sin 40 0.6428° ≈ and ( )cos 40 0.7660° ≈

( ) ( ) [ ]( ) ( ) [ ]

( )

12

12

312 2

Use sin cos sin( ) sin( )

so sin 10 cos 50 sin(60 ) sin( 40 )

0.6428

0.112

A B A B A B= + + −

° ° = × ° + − °

= × −

=

d ( ) ( )cos 10 sin 50° ° = ? given ( )sin 40 0.6428° ≈ and ( )cos 40 0.7660° ≈

( ) ( ) [ ]( ) ( ) [ ]

( )

12

12

312 2

Use cos sin sin( ) sin( )

so cos 10 sin 50 sin(60 ) sin( 40 )

0.6428

0.754

A B A B A B= + − −

° ° = × ° − − °

= × +

=

© Cengage Learning Australia 2014 ISBN 9780170251501 35

Page 36: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

2 a 2 sin (5x) cos (7x) = sin (12x) + sin (–2x) = sin (12x) – sin (2x)

b 8 sin (3x) sin (9x) = 4[cos(–6x) – cos(12x)] = 4 cos (6x) – 4 cos (12x)

c 6 cos (4x) cos (6x) = 3 cos (10x) + 3 cos(–2x) = 3 cos (10x) + 3 cos (2x)

d 4 cos (5x) sin (6x) = 2[sin(11x) – sin(–2x)] = 2 sin (11x) + 2 sin (2x)

e sin (6x) cos (10x) = ( ) ( ) ( ) ( )1 1sin 16 sin 4 sin 16 sin 4

2 2x x x x+ − = −

f 3 cos (10x) cos (14x) = ( ) ( ) ( ) ( )3 3cos 24 cos 4 cos 24 cos 4

2 2x x x x+ − = +

g 5 sin (2x) sin (3x) = ( ) ( ) ( ) ( )5 5cos cos 5 cos cos 5

2 2x x x x− − = −

h 7 cos (9x) cos (11x) = ( ) ( ) ( ) ( )7 7cos 20 cos 2 cos 20 cos 2

2 2x x x x+ − = +

i cos (8x) sin (x) = ( ) ( ) ( ) ( )1 1sin 9 sin 7 sin 9 sin 7

2 2x x x x− − = +

3 a cos cos4 12π π

= ?

( ) ( )2cos cos cos( ) cos( )

1cos cos cos cos4 12 2 4 12 4 12

1 2 4cos cos2 12 12

1 cos cos2 6 3

1 3 12 2 2

3 1

Use

4

x y x y x y= − + +

π π π π π π = × − + + π π = × + π π = × +

= × +

+=

© Cengage Learning Australia 2014 ISBN 9780170251501 36

Page 37: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

b sin cos4 12π π

= ?

( ) ( )2sin cos sin( ) sin( )

1sin cos sin sin4 12 2 4 12 4 12

1 4 2sin sin2 12 12

1 sin sin2 3 6

1 3 12 2 2

3 1

Use

4

x y x y x y= + + −

π π π π π π = × + + − π π = × + π π = × +

= × +

+=

c sin sin12 4π π

= ?

( ) ( )2sin sin cos( ) cos( )

1sin sin cos cos12 4 2 12 4 12 4

1 2 4cos cos2 12 12

1 cos cos2 6 3

1 3 12 2 2

3 1

U

4

se x y x y x y= − − +

π π π π π π = × − − + − π π = × − −π π = × −

= × −

−=

© Cengage Learning Australia 2014 ISBN 9780170251501 37

Page 38: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

d 2sin cos12 4π π

= ?

( ) ( )2sin cos sin( ) sin( )

2sin cos sin sin12 4 12 4 12 4

4 2sin sin12 12

sin sin3 6

3 12 23 1

U e

2

s x y x y x y= + + −

π π π π π π = + + −

π − π = + π π = −

= −

−=

Reasoning and communication

4 Show that ( ) ( ) [ ]12sin cos sin( ) sin( )A B A B A B= + + −

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

1 1sin( ) sin( ) sin cos cos sin sin cos sin cos2 2

1 2sin cos2sin cos

A B A B A B A B A B B A

A B

A B

+ + − = + + −

=

=

5 Show that ( ) ( )2cos sin sin( ) sin( )A B A B A B= + − −

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( )

sin( ) sin( ) sin cos cos sin sin cos sin cos

2cos sin

A B A B A B A B A B B A

A B

+ − − = + − − =

© Cengage Learning Australia 2014 ISBN 9780170251501 38

Page 39: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

6 Show that ( ) ( ) [ ]1sin sin cos( ) cos( )2

g f f g f g= − − +

[ ] ( ) ( )

( ) ( )

( ) ( )

1 1 1 1cos( ) cos( ) 2sin sin2 2 2 2

1 1sin 2 sin 22 2

sin sin

f g f g f g f g f g f g

f g

f g

− − + = − + + + − − =

=

7 Show that ( ) ( )sin( ) sin( ) 2sin cosc d c d c d+ = − −

Using sin (x) – sin (y) = 2 cos sin 2 2

x y x y+ −

( ) ( )sin( ) sin( ) 2sin cosc d c d c d+ − − =

∴ ( ) ( )sin( ) sin( ) 2sin cosc d c d c d+ = − +

8 Show that sin (11x) cos (7x) − sin (8x) cos (4x) = sin (3x) cos (15x).

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ){ }

( ) ( )

( ) ( )

( ) ( )

sin 11 cos 7 sin 8 cos 41 sin 18 sin 4 sin 12 sin 421 sin 18 sin 1221 2sin 15 cos 32sin 15 cos 3

x x x x

x x x x

x x

x x

x x

= + − +

= +

=

=

9 Show that 4 sin (3x) sin (5x) sin (8x) = sin (6x) + sin (10x) − sin (16x).

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

sin 6 sin 10 sin 16

2sin 8 cos( 2 ) 2sin 8 cos 8

2sin 8 cos 2 cos 8

2sin 8 2sin 5 sin 3

4sin 3 sin 5 sin 8

x x x

x x x x

x x x

x x x

x x x

+ −

= − −

= − = =

© Cengage Learning Australia 2014 ISBN 9780170251501 39

Page 40: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

10 If w(t) = sin (5t) + sin (6t) + sin (7t) + sin (8t), then it can be seen in the diagram

below that every sixth wave is larger.

© Cengage Learning Australia 2014 ISBN 9780170251501 40

Page 41: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.07: Sums to products

Concepts and techniques

1 a Express 6 sin (x) + 8 cos (x) in each of the following forms and sketch the

graphs for x = 0 to 2π.

i A sin (x + θ) = Asin (x) cos (θ) + A cos (x) sin (θ)

= 6 sin (x) + 8 cos (x)

So 6 = A cos (θ) and 8 = A sin (θ)

62 + 82 = A2[cos2 (θ)+ sin2 (θ)]

∴62 + 82 = A2

A > 0, A = 10

( )( )

( )

sin 8cos 6

8tan6

0.9273

AA

θ=

θ

θ =

θ =

∴6 sin (x) + 8 cos (x) = 10 sin (x + 0.9273)

© Cengage Learning Australia 2014 ISBN 9780170251501 41

Page 42: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

ii A cos (x + θ) = Acos (x) cos (θ) – A sin (x) sin (θ)

= 8 cos (x) + 6 sin (x)

So 8 = A cos (θ) and 6 = –A sin (θ)

82 + 62 = A2[cos2 (θ)+ sin2 (θ)]

∴82 + 62 = A2

A > 0, A = 10

( )( )

( )

sin 6cos 8

6tan8

0.6435

AA

θ −=

θ

−θ =

θ = −

∴6 sin (x) + 8 cos (x) = 10 cos (x – 0.6435)

© Cengage Learning Australia 2014 ISBN 9780170251501 42

Page 43: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

iii A sin (x − θ) = Asin (x) cos (θ) – A cos (x) sin (θ)

= 6 sin (x) + 8 cos (x)

So 6 = A cos (θ) and 8 = –A sin (θ)

62 + 82 = A2[cos2 (θ)+ sin2 (θ)]

∴62 + 82 = A2

A > 0, A = 10

( )( )

( )

sin 8cos 6

8tan6

0.9273

AA

θ −=

θ

−θ =

θ = −

∴6 sin (x) + 8 cos (x) = 10 sin [x –(–0.9273)]

= 10 sin (x + 0.9273)

© Cengage Learning Australia 2014 ISBN 9780170251501 43

Page 44: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

iv A cos (x − θ) = Acos (x) cos (θ) + A sin (x) sin (θ)

= 8 cos (x) + 6 sin (x)

So 8 = A cos (θ) and 6 = A sin (θ)

82 + 62 = A2[cos2 (θ)+ sin2 (θ)]

∴82 + 62 = A2

A > 0, A = 10

( )( )

( )

sin 6cos 8

6tan8

0.6435

AA

θ=

θ

θ =

θ =

∴6 sin (x) + 8 cos (x) = 10 cos(x – 0.6435)

b The graphs are all the same because they all come from the same the original

function.

© Cengage Learning Australia 2014 ISBN 9780170251501 44

Page 45: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

2 Express ( ) ( )sin 3 cosx x+ in the form A sin (x + θ) and sketch the graph for

x = 0 to 2π.

( ) ( )sin 3 cosx x+ = A sin (x) cos (θ)+ A cos (x) sin (θ)

So 1 = A cos (θ) and 3 = A sin (θ)

12 + 3 2 = A2[cos2 (θ)+ sin2 (θ)]

∴12 + 3 2 = A2

A > 0, A = 2

( )( )

( )

sin 3cos 1

tan 3

3

AA

θ=

θ

θ =

πθ =

∴ ( ) ( )sin 3 cosx x+ = 2 sin 3x π +

© Cengage Learning Australia 2014 ISBN 9780170251501 45

Page 46: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

3 A cos (x − θ) = Acos (x) cos (θ) + A sin (x) sin (θ)

= sin (x) + cos (x)

So 1 = A cos (θ) and 1 = A sin (θ)

12 + 12 = A2[cos2 (θ)+ sin2 (θ)]

∴12 + 12 = A2

A > 0 A = 2

( )( )

( )

sin 1cos 1

tan 1

4

AA

θ=

θ

θ =

πθ =

∴sin (x) + cos (x) = 2 cos 4x π −

© Cengage Learning Australia 2014 ISBN 9780170251501 46

Page 47: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

4 A cos (x + θ) = Acos (x) cos (θ) – A sin (x) sin (θ)

= cos 3 sinx x− +

So –1 = A cos (θ) and – 3 = A sin (θ)

(–1)2 + (– 3 )2 = A2[cos2 (θ)+ sin2 (θ)]

∴ (–1)2 + (– 3 )2 = A2

A > 0, A = 2

( )( )

( )

sin 3cos 1

tan 343

AA

θ −=

θ −

θ =

πθ =

∴ ( ) ( )3 sin cosx x− = 2 cos

43

x π +

© Cengage Learning Australia 2014 ISBN 9780170251501 47

Page 48: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

5 12 sin (x) + 5 cos (x) = 13 sin (x + θ) and tan–1(θ) = 512

so θ = 0.3948

12 sin (x) + 5 cos (x) = 13 sin (x +0.3948)

6 8 sin (x) − 15 cos (x) = 17 sin (x − θ) and tan–1(θ) = 158

so θ = 1.08

8 sin (x) − 15 cos (x) = 17 sin (x −1.08)

© Cengage Learning Australia 2014 ISBN 9780170251501 48

Page 49: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

7 a sin (3x) + sin (7x) = 2 sin (5x) cos(–2x) = 2 sin (5x) cos (2x)

b sin (8x) − sin (2x) = 2 cos (5x) sin (3x)

c cos (5x) + cos (9x) = 2 cos (7x) cos (2x)

d cos (11x) − cos (7x) = 2 sin (9x) sin (–2x) = –2 sin (9x) sin (2x)

e sin (4x) + sin (9x) = 2 sin (6.5x) cos (2.5x)

f sin (2x) − sin (5x) = 2 cos (3.5x) sin(–1.5x) = –2 cos (3.5x) sin (1.5x)

g cos (x) + cos (5x) = 2 cos (3x) cos (2x)

h cos (3x) − cos (11x) = 2 sin (7x) sin (4x)

i sin (x) + sin (2x) = 2 sin (3.5x) cos (0.5x)

8 Using ( ) ( ) ( ) ( )1 12 2cos cos 2cos cosA B A B A B+ = + −

( ) ( )cos 120 cos 60 2cos(90 )cos(30 )

32 02

0

° + ° = ° °

= × ×

=

.

9 Using ( ) ( ) [ ] [ ]1 12 2sin sin 2sin ( ) cos ( )A B A B A B+ = + −

( ) ( )sin 75 sin 15 2sin (45 )cos (30 )

1 3222

3 22 26

2

° + ° = ° °

= × ×

= ×

=

.

10 Using ( ) ( ) [ ] [ ]1 12 2sin sin 2cos ( ) sin ( )A B A B A B− = + −

( ) ( ) ( )sin 135 sin( 45 ) 2cos 45 sin 9012 12

1 222 2

2

° − − ° = ° °

= × ×

= × ×

=

© Cengage Learning Australia 2014 ISBN 9780170251501 49

Page 50: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Reasoning and communication

11 Show that 2 cos (x) cos (y) = cos (x + y) + cos (x − y).

Given

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

cos cos sin sinancos

cos d

cos cos sin sin

x y x y

x y x y

x y

x y

+

= −

= +

by adding we obtain cos (x + y) + cos (x − y). = 2 cos (x) cos (y)

12 Show ( ) ( ) ( ) ( )1 1sin sin 2 sin cos .2 2

A B A B A B + = + −

Given

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

sin cos cos sinands

sin

in

sin cos cos sin

x y

x y

x y x y

x y x y

+ =

−−

+

=

By adding we get

( )sin x y+ + ( )sin x y− = 2 ( ) ( )sin cosx y

Put A = ( )x y+ and B = ( )x y−

then A + B = 2x and A – B = 2y

so and = 2 2

A B A Bx y+ −=

∴ ( )sin x y+ + ( )sin x y− = 2 ( ) ( )sin cosx y becomes

( ) ( ) ( ) ( )1 1sin sin 2 sin cos .2 2

A B A B A B+ = + −

© Cengage Learning Australia 2014 ISBN 9780170251501 50

Page 51: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

13 Show that ( ) ( ) ( ) ( )1 1cos cos 2 cos cos .2 2

A B A B A B + = + −

Given

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

cos cos sin sinancos

cos d

cos cos sin sin

x y x y

x y x y

x y

x y

+

= −

= +

By adding we get

( ) ( ) ( )coco s( ) 2cos cos sx yx y x y+ − =+

Put A = ( )x y+ and B = ( )x y−

then A + B = 2x and A – B = 2y

so and =2 2

A B A Bx y+ −=

∴ ( ) ( ) ( )coco s( ) 2cos cos sx yx y x y+ − =+ becomes

( ) ( ) ( ) ( )1 1cos cos 2 cos cos .2 2

A B A B A B + = + −

14 Both notes are sine waves. At the same intensity (volume) they will add together to

make the sound.

Say one is at f = 110 and the other is at 110 + a

Then sum will be

( ) ( )sin 110 2 sin 110 2 2sin 110 2 cos 22 2a aa t t t t + π + × π = + π × × π

As a decreases, the beats will decrease in frequency (will get further apart) and the

dominant note 110 hz remain essentially unchanged.

© Cengage Learning Australia 2014 ISBN 9780170251501 51

Page 52: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

15 a sin (x + h) − sin (x) = ( ) ( )2cos 0.5 sin 0.5x h h+

b ( ) ( ) ( )2cossin ( ) sin si0 n.5 0.5x x h hh

hx

h++ −

= .

c Given that ( )

0

sinlim 1z

zz→

= , find ( )sind xdx

from the definition of the derivative.

( ) ( ) ( )

( ) ( )

( ) ( )

( )( )

0

0

0

2cos 0.5 0.5

cos 0.5 0.

sinsin lim

sinlim

sinlim

50.5

0.50.5cos

cos 1

c s

5

o

0.

h

h

h

x h hh

x h hh

hx h

d xdx

x

x

h

=

=

=

= ×

+

+

=

+

© Cengage Learning Australia 2014 ISBN 9780170251501 52

Page 53: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

Exercise 9.08: Trigonometric identities

Reasoning and communication

1 Prove the identity 2 2 21 1sin (4 ) sin (2 )cos (2 )8 2

x x x=

( ) ( )

( ) ( )

( ) ( )

22

2 2

2 2

1 1sin (4 ) 2sin 2 cos 28 2

1 4sin 2 cos 222sin 2 cos 2

x x x

x x

x x

=

=

=

2 Show that 2 2 42sin (6 ) 8 sin (3 ) sin (3 )x x x = −

( ) ( )( ) ( )

( ) ( )

22

2 2

2 2

2sin (6 ) 2 2sin 3 cos 3

2 4sin 3 cos 3

8sin 3 cos 3

x x x

x x

x x

= =

=

3 Prove the following identities.

a ( ) ( ) ( ) ( ) 22 22sin 2sin cos sin 1x x x x− − = − −

( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( )

( )

( ) ( )

( )

2 2 2 2

2

2

22

2

2sin 2sin cos 2sin 2sin 1 sin

sin 2sin 1

sin 1

1 1 sin

1 sin

x x x x x x

x x

x

x

x

− − = − − + −

= − − +

= − −

= − − −

= − −

© Cengage Learning Australia 2014 ISBN 9780170251501 53

Page 54: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

b ( )1 cos cos cos cos2 3 2 6 2 6

x xx π π π + = + −

( )

( )

( )

1cos cos cos cos2 6 2 6 2 2 6 2 6 2 6 2 6

1 2cos cos2 6

1 cos cos2 3

1 cos cos2 3

x x x x x x

x

x

x

π π π π π π + − = + − − + + + − π = + π = + π = +

4 Prove the following identities.

a ( ) ( )

1cosec cotx x

=+

[ ]( ) ( )

12

12

1 cos (2 )tan cos sin (2 )

xx x x

+

Take RHS:

[ ]( ) ( )

( ){ }( ) ( )

( ) ( ) ( ){ }( )

( ) ( ) ( )( )

( )[ ]( )

( ) ( )

( )

( )( )

( )

( ) ( )

21122

1 12 2

212

2

sin coscos

1 1 2sin1 cos (2 )tan cos sin (2 ) 2sin cos

2sinsin sin cos

sinsin 1 cos ( )

sin1 cos ( )

1sin sin

11 cos ( )sin

1cos1

sin sin1

cos ec cot

x xx

xxx x x x x

xx x x

xx x

xx

x xx

x

xx x

x x

− −− =+ +

×=

+

=+

=+

= ×+

=+

=+

© Cengage Learning Australia 2014 ISBN 9780170251501 54

Page 55: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

b [ ]21 sin ( ) sin ( )4

x y x y+ + − =

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2sin sin sin 2sin sin 2sin sinx x y x y x y− − +

Take LHS:

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( )

22

2

2 2

1 1sin ( ) sin ( ) sin cos cos sin sin cos cos sin4 4

sin cos

sin cos

x y x y x y x y x y x y

x y

x y

+ + − = + + −

= =

Take RHS:

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ){ }( )

2 2 2 2

22 2

2

sin sin sin 2sin sin 2sin sin

sin 1 sin 2sin 2sin

sin 1 2sin

x x y x y x y

x y y y

x y

− − +

= − − +

= − ( ) ( ) ( )2 2sin 2sin 2siny y y+ − +

( ) ( )( ) ( )

2 2

2 2

sin 1 sin

sin cos

x y

x y

= −

=

∴ [ ]21 sin( ) sin( )4

x y x y+ + − =

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2sin sin sin 2sin sin 2sin sinx x y x y x y− − +

© Cengage Learning Australia 2014 ISBN 9780170251501 55

Page 56: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

c ( ) ( )

1sin cosecx x

=+

( )( ) ( ) ( )

12

2 2 2

sec sin (2 )1 cos cosec cot

x xx x x− + −

Take RHS:

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )( )

( )( ) ( )

( )

( )( ) ( )

( )( )

( ) ( )( )( )

( )( )

( )

( )

( ) ( )

( ) ( )

11 222 2 2

2

4 2

2

2 2

2

2

2

2 2

4 2

2

1cos

cos1sin sin

sin 1 cossin

2sin cossec sin (2 )1 cos cosec cot sin

sin

sinsin sin

sin

sinsin sin 1

sin1

sin sin1sin 1

sin1

1sin sin1

sin cosec

x

xx x

x xx

x xx xx x x x

x

xx x

x

xx x

x

x xx

x

x x

x x

+ −

× ×=

− + − + −

=

=+

=+

= ×+

=+

=+

© Cengage Learning Australia 2014 ISBN 9780170251501 56

Page 57: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

5 1 sin(3 ) cos(2 )3

x x+ in terms of sin only

( )

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ){ } ( )

( ) ( ) ( ) ( ){ } ( )

( ) ( ) ( ) ( )

( )

2

2

2 2

2 2 2

2 2

3

1 sin(2 ) 1 2sin31 sin 2 cos sin cos 2 1 2sin31 2sin cos cos sin 1 2sin 1 2sin31 2sin 1 sin sin 1 2sin 1 2sin3sin 21 1 2sin sin 1 sin

3 3

2 sin 2si3

x x x

x x x x x

x x x x x x

x x x x x

xx x x

x

= + + −

= + + −

= + − + −

= − + − + −

= + − + −

= − − ( ) ( ) ( ) ( )

( ) ( ) ( )

2 3

3 2

sin 2 2n 1 sin sin3 3 3

4 sin 2sin sin 13

xx x x

x x x

+ + + −

= − − + +

6 ( )22sin cos(2 )x x− in terms of ( )cos x only.

[ ]( )

2

2 2

2 2

4 2 2

4 2

2sin ( ) cos (2 ) 1 cos (2 )

1 2cos ( ) 1 2cos ( ) 1

2cos ( ) 2 2cos ( ) 1

4cos ( ) 2cos ( ) 4cos ( ) 24cos ( ) 6cos ( ) 2

x x x

x x

x x

x x xx x

− = − +

= − + − −

= − − = − − +

= − +

7 Prove that sec2 (x) = sec (x) cosec (x) tan (x).

( ) ( ) ( ) ( ) ( )( )( )

( )( )

2

2

sin1 1sec cosec tancos sin cos

1cos

sec

xx x x

x x x

x

x

= × ×

=

=

© Cengage Learning Australia 2014 ISBN 9780170251501 57

Page 58: NELSON SENIOR MATHS SPECIALIST 11mathsbooks.net/Nelson Specialist 11/Nelson...16 Use the graph of y = cos (3x) to sketch the graph of y = sec (3x) from −. π 2 to π 2. 17 Use the

8 Prove that ( ) ( ) ( ) ( )1 1 2 tan sec

cosec 1 cosec 1x x

x x+ =

+ −

( ) ( ) ( ) ( )

( )( )

( )( )

( )( )

( )( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

2

1 1sin sin

1 sin 1 sinsin sin

1 1 1 1cosec 1 cosec 1 1 1

1 1

sin sin1 sin 1 sin

1 sin 1 sinsin

1 sin 1 sin

2sincos

2 tan sec

x x

x x

x x

x x

x xx x

x xx

x x

xx

x x

+ −

+ = ++ − + −

= +

= ++ −

− + + = + −

=

=

.

9 Show that 1 + cos (2x) = 2 cos2 (x).

1 + cos (2x) = ( ) ( )2 21 2cos 1 2cosx x+ − =

10 Prove that ( ) ( ) ( ) ( )cosec coseccot cot

cosec ( )x y

x yx y

+ =+

.

( ) ( )( ) ( )( ) ( ) ( ) ( )

( ) ( )( )( )

( )( )

( ) ( )

cosec cosec sin( )cosec ( ) sin sin

sin cos cos sinsin sin

cos cossin sin

cot cot

x y x yx y x y

x y x yx y

y xy x

y x

+=

+

+=

= +

= +

© Cengage Learning Australia 2014 ISBN 9780170251501 58

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Chapter 9 Review

Multiple choice

1 ( )sin 60° = 32

C

2 ( )cosec 60° =23

A

3 5 1sin sin6 6 2π π π − = =

E

4 ( ) ( ) ( )tan 360 30 tan 330 tan 30° − ° = ° = − ° C

5 cos(75°) =

( ) ( ) ( ) ( )

( )

75 cos 5 30 cos 5 cos 30 sin 5 sin 3

cos( ) (40

1 3 1 12 22

)4 4

23 1 2

2 2 2

2 3 1

4

° = ° + °

= ° ° − ° °

= × − ×

−= ×

−=

A

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6 5sin8π

= ?

( ) ( )

( )

2

2 5 5cos8 4

5sin 08

1 cos 25s

1cos2

cos 2 1 2sin

12

2 12

2 1 22 2

2 2

in8 2

1

2

2

2

4

2 22

A

A

A

× π π

π >

−π = +

=

= −

= −

−=

+=

+=

+=

+=

×

E

7 E cos (5x) cos (9x)

[ ]

[ ]

[ ]

1 cos (5 9 ) cos (5 9 )21 cos (14 ) cos ( 4 )21 cos (14 ) cos (4 )2

x x x x

x x

x x

= + + −

= + −

= +

E

8 sin (12x) − sin (8x)

( ) ( )

12 8 12 82 cos sin2 2

2cos 10 sin 2

x x x x

x x

+ − = =

B

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Short answer

9 a ( )sec(2 ) secπ − θ = θ

b ( )cosec( ) cos ecπ − θ = θ

c ( )cot(2 ) cotπ − θ = − θ

d ( )sec( ) secπ + θ = − θ

e ( )cot( ) cotπ + θ = θ

f ( )cosec(2 ) cos ecπ − θ = − θ

10 a ( )sec cosec2π − θ = θ

b ( )3cosec cosec cosec s ec2 2 2π π π − θ = π + − θ = − − θ = − θ

c ( )cot tan2π − θ = θ

d ( )3sec sec sec cosec2 2 2π π π + θ = π + + θ = − + θ = θ

e ( )3cot cot cot tan2 2 2π π π − θ = π + − θ = − θ = θ

f ( )cosec sec2π − θ = θ

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11 ( )3cosecy x= for 0 2 ,x x< < π ≠ π

12 ( )2sec 1y x= − for 30 2 , ,2 2

x x π π≤ ≤ π ≠

13 Prove the ( ) ( ) ( )2 3sec 1 tan cotx x x− =

( ) ( )( )

( )( )( )

( ) ( )( )

( ) ( )

22

2

2

2

2

2

3

1sec 1 1cos

1 coscos

sincos

tantan

tan

tan cot

xx

xx

xx

xx

x

x x

− = −

−=

=

= ×

=

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14 cos cos12 3 4π π π = −

cos cos sin sin3 4 3 4

1 1 3 12 22 21 3 22 2 22 6

4

π π π π = +

= × + ×

+= ×

+=

15 tan tan12 3 4π π π = −

( )

( )

2

tan tan3 4

1 tan tan3 4

3 1 1 31 3 1 3

1 3

1 31 2 3 3

24 2 3

22 3

π π − =

π π +

− −= ×

+ −

−= −

− += −

−−

=

= −

16 7sin sin12 3 4π π π = +

sin cos cos sin3 4 3 4

3 1 1 12 22 23 1 2

2 2 26 2

4

π π π π = +

= × + ×

+= ×

+=

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Application

17 ( ) ( ) ( )sin 4 2sin 2 cos 2x x x=

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2

3 3

2 2sin cos cos sin

4sin cos 4sin cos

x x x x

x x x x

= − = −

18 Show that ( ) ( ) [ ]12sin sin cos( ) cos( )A B A B A B= − − + .

Given ( ) ( ) ( ) ( )cos( ) cos cos sin sinA B A B A B− = +

and ( ) ( ) ( ) ( )cos( ) cos cos sin sinA B A B A B+ = −

then cos (A –B) – cos (A + B) = ( ) ( ) ( ) ( )cos cos sin sinA B A B+

– ( ) ( ) ( ) ( )cos cos sin sinA B A B−

i.e. cos(A –B) – cos (A + B) = 2 ( ) ( )sin sinA B

∴ ( ) ( ) [ ]12sin sin cos( ) cos( )A B A B A B= − − +

19 Show that ( ) ( )

( )( ) ( )( ) ( )

2

2

2 tan cos 1cos seccot 2 cos 1 tan

x xx xx x x

++ = −

( ) ( )( ) ( )

( )( )

( )( )

( ) ( ) ( )( ) ( )

( )

2 2

22

2 tan cos 1 cos 12 tancos1 tancos 1 tan

tan 2 cos sec

cos seccot 2

x x xxxxx x

x x x

x xx

+ + = ×− −

= × + +

=

© Cengage Learning Australia 2014 ISBN 9780170251501 64

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20 Show that ( ) ( ) ( ) ( ) ( ) 231 cos

4sin cos 2sin cos8

xx x x x

− = −

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( )

( ) ( ) ( )

( )

( )

2 223 2

2 2

2

2

2 2

4sin cos 2sin cos 2sin cos 2cos 1

sin 2 cos 2

2sin 2 cos 24

sin 44

1 2sin 4 but cos 2 =1 2sin81 1 cos 881 cos 8

8

x x x x x x x

x x

x x

x

x A A

x

x

− = −

=

=

=

= × −

= × −

−=

21 Show that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2cos tan sin 3 sin cos 2 sin cos sin 2x x x x x x x x= +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

2sin cos 2 sin cos sin 2 sin sin cos 2 cos sin 2

sin sin( 2 )

sin sin 3

cossin sin 3

cos

cos tan sin 3

x x x x x x x x x x

x x x

x x

xx x

x

x x x

+ = + = +

=

= ×

=

22 Rewrite the expression ( ) ( ) ( )3sin(2 )sin cos cos cos(2 )x x x x x+ in terms of

( )cos x only.

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

2

2 2 2

3sin(2 )sin cos cos cos(2 )

3 2sin cos sin cos cos 2cos 1

6cos ( ) 1 cos cos 2cos 1

x x x x x

x x x x x x

x x x x

+

= + − = − + −

© Cengage Learning Australia 2014 ISBN 9780170251501 65