Higher%20Unit%202

1
Higher Unit 2
Trigonometry identities of the form sin(AB)
Double Angle formulae
Trigonometric Equations
Radians Trig Basics
More Trigonometric Equations
Exam Type Questions
2
Trig Identities
Supplied on a formula sheet !!
The following relationships are always true for
two angles A and B.
1a. sin(A B) sinAcosB cosAsinB
1b. sin(A - B) sinAcosB - cosAsinB
2a. cos(A B) cosAcosB sinAsinB
2b. cos(A - B) cosAcosB sinAsinB
Quite tricky to prove but some of following
examples should show that they do work!!
3
Trig Identities
Examples 1
(1) Expand cos(U V).
(use formula 2b )
cos(U V) cosUcosV sinUsinV
(2) Simplify sinfcosg - cosfsing
(use formula 1b )
sinfcosg - cosfsing sin(f g)
(3) Simplify cos8 ? sin? sin8 ? cos ?
(use formula 1a )
cos8 ? sin ? sin8 ? cos ?
sin(8 ? ?)
sin9 ?
4
Trig Identities
Example 2
By taking A 60 and B 30, prove the
identity for cos(A B).
NB cos(A B) cosAcosB sinAsinB
cos30
LHS cos(60 30 )
?3/2
RHS cos60cos30 sin60sin30
( ½ X ?3/2 ) (?3/2 X ½)
?3/4 ?3/4
?3/2
Hence LHS RHS !!
5
Trig Identities
Example 3
Prove that sin15 ¼(?6 - ?2)
sin15 sin(45 30)
sin45cos30 - cos45sin30
(1/?2 X ?3/2 ) - (1/?2 X ½)
(?3/2?2 - 1/2?2)
(?3 - 1) 2?2
X ?2 ?2
(?6 - ?2) 4

¼(?6 - ?2)
6
Trig Identities
NAB type Question
Example 4
y
41
3
x
?
?
4
40
Show that cos(? - ?) 187/205
Triangle1
Triangle2
If missing side y
If missing side x
Then x2 412 402 81
Then y2 42 32 25
So x 9
So y 5
sin? 9/41 and cos? 40/41
sin ? 3/5 and cos? 4/5
7
Trig Identities
sin? 9/41 and cos? 40/41
sin ? 3/5 and cos? 4/5
cos(? - ?) cos?cos? sin?sin?
(40/41 X 4/5) (9/41 X 3/5 )
160/205 27/205
187/205
Remember this is a NAB type Question
8
NAB type Question
Trig Identities
Example 5
Solve sinx?cos30? cosx?sin30? -0.966
where 0o lt x lt 360o
ALWAYS work out Quad 1 first
By rule 1a sinx?cos30? cosx?sin30?
sin(x 30)?
sin(x 30)? -0.966
Quad 3 and Quad 4
sin-1 0.966 75?
Quad 3 angle 180o 75o
Quad 4 angle 360o 75o
x 30o 285o
x 30o 255o
x 225o
x 255o
9
Trig Identities
Example 6
Solve sin5 ? cos3 ? - cos5 ? sin3 ? ?3/2
where 0 lt ? lt ?
sin(5? - 3?)
sin2?
By rule 1b. sin5? cos3? - cos5? sin3?
sin2? ?3/2
Quad 1 and Quad 2
sin-1 ?3/2 ?/3
Repeats every ?
Quad 1 angle ?/3
Quad 2 angle ? - ?/3
In this example repeats lie out with limits
2 ? ?/3
2 ? 2?/3
? ?/3
? ?/6
10
Trig Identities
Example 7
Find the value of x that minimises the
expression cosx?cos32? sinx?sin32?
Using rule 2(b) we get
cosx?cos32? sinx?sin32? cos(x 32)?
cos graph is roller-coaster
min value is -1 when angle 180?
ie x 32o 180o
ie x 212o
11
Paper 1 type questions
Trig Identities
Example 8
Simplify sin(? - ?/3) cos(? ?/6) cos(?/2
- ?)
sin(? - ?/3) cos(? ?/6) cos(?/2 - ?)
sin ? cos?/3 cos ? sin?/3
cos ? cos?/6 sin ? sin?/6 cos?/2
cos ? sin?/2 sin ?
1/2 sin ? ?3/2cos ? ?3/2 cos ? 1/2sin ?
0 x cos ? 1 X sin ?

sin ?
12
Paper 1 type questions
Trig Identities
Example 9
Prove that (sinA cosB)2 (cosA - sinB)2
2(1 sin(A - B))
LHS (sinA cosB)2 (cosA - sinB)2
sin2A 2sinAcosB cos2B cos2A 2cosAsinB
sin2B
(sin2A cos2A) (sin2B cos2B) 2sinAcosB
- 2cosAsinB
1 1 2(sinAcosB - cosAsinB)
2 2sin(A B)
2(1 sin(A B))
RHS
13
Double Angle Formulae
14
Double Angle formulae
Mixed Examples
Substitute form the tan (sin/cos) equation
ve because A is acute
3-4-5 triangle !
Similarly
A is greater than 45 degrees hence 2A is
greater than 90 degrees.
15
Double Angle formulae
16
Double Angle formulae
17
Double Angle formulae
18
Trigonometric Equations
Double angle formulae (like cos2A or sin2A) often
occur in trig equations. We can solve these
equations by substituting the expressions derived
in the previous sections.
Rules for solving equations
sin2A 2sinAcosA when replacing sin2Aequation
cos2A 2cos2A 1 if cosA is also in the
equation
cos2A 1 2sin2A if sinA is also in the equation
19
Trigonometric Equations
cos2x and sin x, so substitute 1-2sin2x
20
Trigonometric Equations
cos 2x and cos x, so substitute 2cos2 -1
21
Trigonometric Equations
22
Trigonometric Equations
Three problems concerning this graph follow.
23
Trigonometric Equations
The max min values of sinbx are 1 and -1 resp.
The max min values of asinbx are 3 and -3 resp.
f(x) goes through 2 complete cycles from 0 360o
The max min values of csinx are 2 and -2 resp.
24
Trigonometric Equations
From the previous problem we now have
Hence, the equation to solve is
Expand sin 2x
Divide both sides by 2
Spot the common factor in the terms?
Is satisfied by all values of x for which
25
Trigonometric Equations
From the previous problem we have
Hence
26
Radian Measurements
Reminders
i) Radians
Converting between degrees and radians
27
Degree Measurements
Equilateral triangle
ii) Exact Values
45o right-angled triangle
28
Radians / Degrees
degrees 0o 30o 45o 60o 90o
radians
sin
cos
tan
0
0
1
1
0
0
1
What is the exact value of sin 240o ?
Example
29
Sine Graph
Period 360o
Amplitude 1
30
Cosine Graph
Period 360o
Amplitude 1
31
Tan Graph
Period 180o
Amplitude cannot be found for tan function
32
Solving Trigonometric Equations
Example
Step 2 consider what solutions are expected
Step 1 Re-Arrange
33
Solving Trigonometric Equations
cos 3x is positive so solutions
in the first and fourth
quadrants
x 3
x 3
34
Solving Trigonometric Equations
Step 3 Solve the equation
cos wave repeats every 360o
1st quad
4th quad
3x 60o
300o
420o 660o 780o 1020o
x 20o
100o
140o
220o
260o
340o
35
Solving Trigonometric Equations
Graphical solution for
36
Solving Trigonometric Equations
Example
Step 2 consider what solutions are expected
Step 1 Re-Arrange
sin 6t is negative so solutions in the third and
fourth quadrants
x 6
x 6
37
Solving Trigonometric Equations
Step 3 Solve the equation
sin wave repeats every 360o
3rd quad
4th quad
6t 225o
315o
585o 675o 945o 1035o
x 39.1o
52.5o
97.5o
112.5o
157.5o
172.5o
38
Solving Trigonometric Equations
Graphical solution for
39
The solution is to be in radians but work in
degrees and convert at the end.
Solving Trigonometric Equations
Example
Step 2 consider what solutions are expected
Step 1 Re-Arrange
(2x 60o ) sin-1(1/2)
x 2
x 2
40
Solving Trigonometric Equations
Step 3 Solve the equation
sin wave repeats every 360o
1st quad
2nd quad
2x 90o
210o
450o 570o
x 45o
105o
225o
285o
41
Solving Trigonometric Equations
Graphical solution for
42
The solution is to be in radians but work in
degrees and convert at the end.
Solving Trigonometric Equations
Harder Example
Step 2 consider what solutions are expected
Step 1 Re-Arrange
2 solutions 1st and 3rd quads
2 solutions 2nd and 4th quads
43
Solving Trigonometric Equations
Step 3 Solve the equation
tan wave repeats every 180o
1st quad
2nd quad
x 60o
120o
240o 300o
44
Solving Trigonometric Equations
Graphical solution for
45
Solving Trigonometric Equations
Harder Example
Step 2 Consider what solutions are expected
Step 1 Re-Arrange
One solution
Two solutions
46
Solving Trigonometric Equations
Step 3 Solve the equation
One solution
Two solutions
1stquad
2nd quad
90o
160.5o
x 19.5o
Overall solution x 19.5o , 90o and 160.5o
47
Solving Trigonometric Equations
Graphical solution for
48
The solution is to be in radians but work in
degrees and convert at the end.
Solving Trigonometric Equations
Harder Example
Step 2 Consider what solutions are expected
Step 1 Re-Arrange
One solution
Two solutions
49
Solving Trigonometric Equations
Step 3 Solve the equation
One solution
Two solutions
1stquad
3rd quad
180o
306.9o
x 53.1o
Overall solution in radians x 0.93 , p and
5.35
50
Solving Trigonometric Equations
Graphical solution for
51
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Compound Angles
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Using Compound angle formula for
Exact values
Solving equations
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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
Pythagoras
Write down values for cos p and sin p
Expand sin (2p)
Expand cos (2p)
Use m tan (2p)
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In triangle ABC show that the exact value of
Use Pythagoras
Write down values for sin a, cos a, sin b, cos b
Expand sin (a b)
Substitute values
Simplify
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56
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Using triangle PQR, as shown, find the exact
value of cos 2x
Use Pythagoras
Write down values for cos x and sin x
Expand cos 2x
Substitute values
Simplify
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On the co-ordinate diagram shown, A is the point
(6, 8) and B is the point (12, -5). Angle AOC p
and angle COB q Find the exact value of sin
(p q).
Mark up triangles
Use Pythagoras
Write down values for sin p, cos p, sin q, cos q
Expand sin (p q)
Substitute values
Simplify
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58
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Draw triangles
Use Pythagoras
Hypotenuses are 5 and 13 respectively
Write down sin A, cos A, sin B, cos B
Expand sin 2A
Expand cos 2A
Expand sin (2A B)
Substitute
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59
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If x is an acute angle such that show
that the exact value of
5
Draw triangle
Use Pythagoras
Hypotenuse is 5
Write down sin x and cos x
Expand sin (x 30)
Substitute
Simplify
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Table of exact values
60
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Use Pythagoras
Write down sin x, cos x, sin y, cos y.
Expand cos (x y)
Substitute
Simplify
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61
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The framework of a childs swing has
dimensions as shown in the diagram. Find the
exact value of sin x
Draw triangle
Use Pythagoras
Draw in perpendicular
Use fact that sin x sin ( ½ x ½ x)
Write down sin ½ x and cos ½ x
Expand sin ( ½ x ½ x)
Substitute
Simplify
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Table of exact values
62
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Given that
find the exact value of
Draw triangle
Use Pythagoras
Write down values for cos a and sin a
Expand sin 2a
Substitute values
Simplify
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63
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Find algebraically the exact value of
Expand sin (q 120)
Expand cos (q 150)
Use table of exact values
Combine and substitute
Simplify
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Table of exact values
64
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If
find the exact value of a) b)
3
Draw triangle
Use Pythagoras
Opposite side 3
Write down values for cos q and sin q
Expand sin 2q
Expand sin 4q (4q 2q 2q)
Expand cos 2q
Find sin 4q
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65
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For acute angles P and Q
Show that the exact value of
Draw triangles
Use Pythagoras
Adjacent sides are 5 and 4 respectively
Write down sin P, cos P, sin Q, cos Q
Expand sin (P Q)
Substitute
Simplify
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Using Compound angle formula for
Solving Equations
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Replace cos 2x with
Determine quadrants
Substitute
Simplify
Factorise
Hence
Discard
Find acute x
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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 graph at points A and B. Find
the co-ordinates of B.
Equation
Determine quadrants
Solve simultaneously
Rearrange
Check range
Find acute 2x
Deduce 2x
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Table of exact values
70
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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
Determine x
1st expression
2nd expression
Determine quadrants
Form equation
Replace sin 2x
Rearrange
Common factor
Hence
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Table of exact values
71
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• Functions
  are defined on a
  suitable set of real numbers
• Find expressions for i) f(h(x)) ii)
  g(h(x))
• i) Show that
  ii) Find a similar
  expression for g(h(x))
• iii) Hence solve the equation

Simplifies to
1st expression
Rearrange
2nd expression
acute x
Simplify 1st expr.
Use exact values
Determine quadrants
Similarly for 2nd expr.
Form Eqn.
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Table of exact values
72
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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) to write
down the co-ordinates of the point P.
Replace sin 2x
Solutions for where graphs cross
Common factor
Hence
By inspection (P)
Determine x
Find y value
Coords, P
Determine quadrants for sin x
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Table of exact values
73
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Solve the equation
for 0 x 360
Replace cos 2x with
Determine quadrants
Substitute
Simplify
Factorise
Hence
Find acute x
Solutions are x 60, 132, 228 and 300
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Table of exact values
74
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Solve the equation
for 0 x 2?
Rearrange
Find acute x
Note range
Solutions are
Determine quadrants
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Table of exact values
75
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a) Write the equation cos 2q 8 cos q 9 0
in terms of cos q and show that for cos
q it has equal roots. b) Show that there are
no real roots for q
Try to solve
Replace cos 2q with
Rearrange
No solution
Divide by 2
Hence there are no real solutions for q
Factorise
Equal roots for cos q
Deduction
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76
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Solve algebraically, the equation sin 2x sin x
0, 0 ? x ? 360
Determine quadrants for cos x
Replace sin 2x
Common factor
Hence
Determine x
x 0, 120, 240, 360
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Table of exact values
77
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Find the exact solutions of 4sin2 x 1, 0
? x ? 2?
Rearrange
Take square roots
Find acute x
and from the square root requires all 4
quadrants
Determine quadrants for sin x
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Table of exact values
78
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Solve the equation
for 0 x 360
Replace cos 2x with
Determine quadrants
Substitute
Simplify
Factorise
Hence
Find acute x
Solutions are x 60, 180 and 300
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Table of exact values
79
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Higher
Solve algebraically, the equation
for 0
x 360
Replace cos 2x with
Determine quadrants
Substitute
Simplify
Factorise
Hence
Discard above
Find acute x
Solutions are x 60 and 300
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Table of exact values
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Table of exact values
30 45 60

sin
cos
tan 1
Return
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