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Dec 15, 2015
Maths4Scotland Higher
Compound Angles
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Using Compound angle formula for
Exact values
Solving equations
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Maths4Scotland Higher
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
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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
Maths4Scotland Higher
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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Draw triangle Use Pythagoras
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
Table of exact values
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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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)
Table of exact values
Expand cos (+150)
cos 150 cos cos150 sin sin150
Simplify 1 3 3 1
2 2 2 2sin sin cos cos sin
0
Maths4Scotland Higher
Hint
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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
Draw triangle Use Pythagoras
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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
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Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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°
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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.
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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Solve the equation for 0 ≤ x ≤ 360°3cos(2 ) cos( ) 1x x
Replace cos 2x with 2cos 2 2cos 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
Table of exact values
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
Maths4Scotland Higher
Hint
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Solve the equation for 0 ≤ x ≤ 2 62sin 2 1x
Rearrange
Find acute x 62
6acute x
Determine quadrantsAS
CT
Table of exact values
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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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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
Maths4Scotland Higher
Hint
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Solve the equation for 0 ≤ x ≤ 360°cos 2 cos 0x x
Replace cos 2x with 2cos 2 2cos 1x x
Substitute 22cos 1 cos 0x x
Simplify
Factorise 2cos 1 cos 1 0x x
Hence1
2cos x
Find acute x 60acute x
Determine quadrants
AS
CTcos 1x
180x
Table of exact values
1
2cos x
60
300
x
x
Solutions are: x= 60°, 180° and 300°
60acute x 22cos cos 1 0x x
Maths4Scotland Higher
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Solve algebraically, the equation for 0 ≤ x ≤ 360°cos 2 5cos 2 0x x
Replace cos 2x with 2cos 2 2cos 1x x
Substitute 22cos 1 5cos 2 0x x
Simplify 22cos 5cos 3 0x x
Factorise 2cos 1 cos 3 0x x
Hence1
2cos x
Find acute x 60acute x
Determine quadrants
cos 3x
Table of exact values
AS
CT
1
2cos x
60
300
x
x
Solutions are: x= 60° and 300°
60acute x
Discard above
Maths4Scotland Higher
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