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Compound Angles

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Compound Angles

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Using Compound angle formula for

Exact values

Solving equations

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Hint

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A is the point (8, 4). The line OA is inclined at an angle p radians to the x-axis a) Find the exact values of: i) sin (2p) ii) cos (2p)

The line OB is inclined at an angle 2p radians to the x-axis. b) Write down the exact value of the gradient of OB.

Draw triangle Pythagoras80

Write down values for cos p and sin p8 4

cos sin80 80

p p

Expand sin (2p) sin 2 2sin cosp p p 4 8 64 42

80 580 80

Expand cos (2p) 2 2cos 2 cos sinp p p 2 28 4

80 80

64 16 3

80 5

Use m = tan (2p)sin 2

tan 2cos 2

pp

p 4 3 4

5 5 3

8

4p


Hint


In triangle ABC show that the exact value of

Use Pythagoras

Write down values forsin a, cos a, sin b, cos b

1 1 1 3sin cos sin cos

2 2 10 10a a b b

Expand sin (a + b) sin( ) sin cos cos sina b a b a b

is2

sin( )5

a b

2 10AC CB

2 10

Substitute values1 3 1 1

2 10 2 10sin( )a b

Simplify3 1

20 20sin( )a b 4

20

4 4 2

4 5 2 5 5


Hint


Using triangle PQR, as shown, find theexact value of cos 2x

Use Pythagoras

Write down values forcos x and sin x

2 7cos sin

11 11x x

Expand cos 2x2 2cos 2 cos sinx x x

11PR

11

Substitute values 222 7

11 11cos 2x

Simplify4 7

cos 211 11

x 3

11


Hint


On the co-ordinate diagram shown, A is the point (6, 8) andB is the point (12, -5). Angle AOC = p and angle COB = q Find the exact value of sin (p + q).

Use Pythagoras

Write down values forsin p, cos p, sin q, cos q

8 6 5 12

10 10 13 13sin , cos , sin , cosp p q q

Expand sin (p + q) sin ( ) sin cos cos sinp q p q p q

10 13OA OB

Substitute values

Simplify 126 63

130 65

6

8

512

10

13

Mark up triangles

8 12 6 5

10 13 10 13sin ( )p q

96 30

130 130sin ( )p q


Hint


Draw triangles Use Pythagoras

Expand sin 2A sin 2 2sin cosA A A

A and B are acute angles such that and .

Find the exact value of

a) b) c)

3

4tan A 5

12tan B

sin 2A cos 2A sin(2 )A B4

3A

12

5B

Hypotenuses are 5 and 13 respectively

5 13

Write down sin A, cos A, sin B, cos B 3 4 5 12

, , ,5 5 13 13

sin cos sin cosA A B B

3 4 24

5 5 25sin 2 2A

Expand cos 2A 2 2cos 2 cos sinA A A 2 2 16 9 74 3

25 25 255 5cos 2A

Expand sin (2A + B) sin 2 sin 2 cos cos 2 sinA B A B A B

Substitute 24 12 7 5 323sin 2

25 13 25 13 325A B


Hint


Draw triangle Use Pythagoras

Expand sin (x + 30) sin( 30) sin cos30 cos sin 30x x x

If x° is an acute angle such that

show that the exact value of

4

3tan x

4 3 3sin( 30) is

10x

3

4

x

Hypotenuse is 5

5

Write down sin x and cos x4 3

,5 5

sin cosx x

Substitute

Simplify

Table of exact values

4 3 3 1sin( 30)

5 2 5 2x

4 3 3sin( 30)

10 10x 4 3 3

10


Hint


Use Pythagoras

Expand cos (x + y) cos( ) cos cos sin sinx y x y x y

Write downsin x, cos x, sin y, cos y.

3 4 24 5, , ,

5 5 7 7sin cos sin cosx x y y

Substitute

Simplify20 3 4 6

35

The diagram shows two right angled trianglesABD and BCD with AB = 7 cm, BC = 4 cm and CD = 3 cm. Angle DBC = x° and angle ABD is y°.

Show that the exact value of 20 6 6

cos( )35

x y is

5, 24BD AD

24

5

4 5 3 24cos( )

5 7 5 7x y

20 3 24cos( )

35 35x y

20 6 6

35


Hint



2 5

3 32 2sin , cosx x

The framework of a child’s swing has dimensionsas shown in the diagram. Find the exact value of sin x°

Write down sin ½ x and cos ½ x

5h

Substitute

Simplify


3 3

4

xDraw in perpendicular

2

2

x

h5Use fact that sin x = sin ( ½ x + ½ x)

Expand sin ( ½ x + ½ x) 2 2 2 2 2 22 2sin sin cos sin cos 2sin cosx x x x x xx x

2 5

3 32 2sin 2x x

4 5sin

9x


Hint


Given that

find the exact value of

Write down values forcos a and sin a

3 11cos sin

20 20a a

Expand sin 2a sin 2 2 sin cosa a a

20

Substitute values11 3

sin 2 220 20

a

Simplify

11tan , 0

3 2

3a

11sin 2

Draw triangle Use Pythagoras hypotenuse 20

6 11sin 2

20a

3 11

10


Hint


Find algebraically the exact value of

1 3cos 120 cos 60 cos 150 cos30

2 2

3 1sin 120 sin 60 sin 150 sin 30

2 2

Expand sin (+120)

sin 120 sin cos120 cos sin120

Use table of exact values

1 3 3 1

2 2 2 2sin sin . cos . cos . sin . Combine and substitute

sin sin 120 cos( 150)


Expand cos (+150)

cos 150 cos cos150 sin sin150

Simplify 1 3 3 1

2 2 2 2sin sin cos cos sin

0


Hint


If find the exact value of

a) b)

Write down values forcos and sin

4 3cos sin

5 5

Expand sin 2 sin 2 2 sin cos


4cos , 0

5 2

5

4

3

Opposite side = 3

3 4 242

5 5 25

Expand sin 4 (4 = 2 + 2) sin 4 2 sin 2 cos 2

Expand cos 2 2 2cos 2 cos sin 16 9 7

25 25 25

Find sin 424 7

sin 4 225 25

336

625

sin 2 sin 4


Hint


Draw triangles Use Pythagoras

Expand sin (P + Q) sin sin cos cos sinP Q P Q P Q

For acute angles P and Q

Show that the exact value of12

13

P

53

Q

Adjacent sides are 5 and 4 respectively

5 4

Write down sin P, cos P, sin Q, cos Q 12 5 3 4

, , ,13 13 5 5

sin cos sin cosP P Q Q

Substitute

12 3and

13 5sin sinP Q

63

65sin ( )P Q

12 4 5 3sin

13 5 13 5P Q

Simplify 48 15sin

65 65P Q 63

65


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Solving Equations

Using Compound angle formula for

Continue


Hint


Solve the equation for 0 ≤ x ≤ correct to 2 decimal places 3cos(2 ) 10cos( ) 1 0x x

Replace cos 2x with 2cos 2 2cos 1x x

Substitute 23 2cos 1 10cos 1 0x x

Simplify 26cos 10cos 4 0x x 23cos 5cos 2 0x x

Factorise 3cos 1 cos 2 0x x

Hence 1

3cos

cos 2

x

x

Discard

Find acute x 1.23acute radx

Determine quadrants

AS

CT

1.23 2 1.23or radsx

1.23

5.05

rads

rads

x

x


Hint

Previous NextQuitQuitTable of exact values

Solve simultaneously 2cos 2 3x

Rearrange 3

2cos 2x

0 0 2 2x x

Find acute 2x 62acute x

Determine quadrants

AS

CT

6 6

6 6 6 62 or radsx

5 7

12 12orx

The diagram shows the graph of a cosine function from 0 to .

a) State the equation of the graph.

b) The line with equation y = -3 intersects this graphat points A and B. Find the co-ordinates of B.

Equation 2cos 2y x

Check range

7

12, 3isB B Deduce 2x

Functions f and g are defined on suitable domains by f(x) = sin (x) and g(x) = 2x a) Find expressions for:

i) f(g(x)) ii) g(f(x)) b) Solve 2 f(g(x)) = g(f(x)) for 0 x 360°


Hint


2nd expression

Form equation 2sin 2 2sinx x

Rearrange

Determinequadrants

AS

CT60 , 300x

1st expression ( ( )) (2 ) sin 2f g x f x x

Common factor

( ( )) (sin ) 2sing f x g x x

Replace sin 2x 2sin cos sinx x x

sin 2 sinx x

2sin cos sin 0x x x

sin 2cos 1 0x x

Hence1

or2

sin 0 2cos 1 0 cosx x x

Determine x

sin 0 0 , 360x x

1

2cos 60acutex x

0 , 60 , 300 , 360x

Functions are defined on a suitable set of real numbers

a) Find expressions for i) f(h(x)) ii) g(h(x))

b) i) Show that ii) Find a similar expression for g(h(x))

iii) Hence solve the equation


Hint


2nd expression

Simplify 1st expr.

Similarly for 2nd expr.

Determinequadrants

AS

CT3,

4 4x

1st expression 4 4( ( )) sinf h x f x x

Use exact values

and4

( ) sin , ( ) cos ( )f x x g x x h x x

1 1( ( )) sin cos

2 2f h x x x

for( ( )) ( ( )) 1 0 2f h x g h x x

4 4( ( )) cosg h x g x x

4 4( ( )) sin cos cos sinf h x x x

1 1

2 2( ( )) sin cosf h x x x

4 4( ( )) cos cos sin sing h x x x

1 1

2 2( ( )) cos sing h x x x

Form Eqn. ( ( )) ( ( )) 1f h x g h x

2

2sin 1x Simplifies to

2 2 1

2 2 2 2sin x Rearrange:

acute x 4acute x

a) Solve the equation sin 2x - cos x = 0 in the interval 0 x 180°b) The diagram shows parts of two trigonometric graphs,

y = sin 2x and y = cos x. Use your solutions in (a) towrite down the co-ordinates of the point P.


Hint


Determine quadrantsfor sin x

AS

CT

30 , 150x

Common factor

Replace sin 2x 2sin cos cos 0x x x

cos 2sin 1 0x x

Hence1

or2

cos 0 2sin 1 0 sinx x x

Determine x cos 0 90 , ( 270 )out of rangex x 1

2sin 30acutex x

30 , 90 , 150x

Solutions for where graphs cross

150x By inspection (P)

cos150y Find y value3

2y

Coords, P

3

2150 ,P


Hint


Solve the equation for 0 ≤ x ≤ 360°3cos(2 ) cos( ) 1x x


Substitute 23 2cos 1 cos 1x x

Simplify 26cos cos 2 0x x

Factorise 3cos 2 2cos 1 0x x

Hence2

3cos x

Find acute x 48acute x

Determine quadrants

AS

CT1

2cos x

60acute x


2

3cos x

AS

CT

1

2cos x

132

228

x

x

60

300

x

x

Solutions are: x= 60°, 132°, 228° and 300°

48acute x 60acute x


Hint


Solve the equation for 0 ≤ x ≤ 2 62sin 2 1x

Rearrange

Find acute x 62

6acute x

Determine quadrantsAS

CT


Solutions are:

6

1sin 2

2x

62

6x 6

52

6x

Note range 0 2 0 2 4x x

and for range 2 2 4x

6

132

6x 6

172

6x

7 3, , ,

6 2 6 2x

for range 0 2 2x


Hint


a) Write the equation cos 2 + 8 cos + 9 = 0 in terms of cos and show that for cos it has equal roots.

b) Show that there are no real roots for

Rearrange

Divide by 2

Deduction

Factorise cos 2 cos 2 0

Replace cos 2 with 2cos 2 2cos 1

22cos 8cos 8 0

2cos 4cos 4 0

Equal roots for cos

Try to solve:

cos 2 0

cos 2

Hence there are no real solutions for

No solution

Solve algebraically, the equation sin 2x + sin x = 0, 0 x 360


Hint


Determine quadrantsfor cos x

AS

CT

120 , 240x

Common factor

Replace sin 2x 2sin cos sin 0x x x

sin 2cos 1 0x x

Hence1

or2

sin 0

2cos 1 0 cos

x

x x

Determine x sin 0 0 , 360x x

1

2cos 60acutex x

x = 0°, 120°, 240°, 360°

Find the exact solutions of 4sin2 x = 1, 0 x 2


Hint


Determine quadrants for sin x

AS

CT

Take square roots

Rearrange 2 1

4sin x

1

2sin x

Find acute x6

acute x

+ and – from the square root requires all 4 quadrants

5 7 11, , ,

6 6 6 6x


Hint


Solve the equation for 0 ≤ x ≤ 360°cos 2 cos 0x x


Substitute 22cos 1 cos 0x x

Simplify


Hence1

2cos x


Determine quadrants

AS

CTcos 1x

180x


1

2cos x

60

300

x

x

Solutions are: x= 60°, 180° and 300°

60acute x 22cos cos 1 0x x


Hint


Solve algebraically, the equation for 0 ≤ x ≤ 360°cos 2 5cos 2 0x x


Substitute 22cos 1 5cos 2 0x x

Simplify 22cos 5cos 3 0x x


Hence1

2cos x


Determine quadrants

cos 3x


AS

CT

1

2cos x

60

300

x

x

Solutions are: x= 60° and 300°

60acute x

Discard above


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30° 45° 60°

sin

cos

tan 1

6

4

3

1

2

1

23

2

3

2

1

21

21

3 3


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