1. Higher Unit 2 Trigonometry identities of the form sin(A+B)
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 (1) Expand cos(U – V). (use formula 2b )
Examples 1
(1) Expand cos(U – V).
(use formula 2b )
cos(U – V) = cosUcosV + sinUsinV
(2) Simplify sinf°cosg° - cosf°sing°
(use formula 1b )
sinf°cosg° - cosf°sing° = sin(f – g)°
(3) Simplify cos8 θ sinθ + sin8 θ cos θ
(use formula 1a )
cos8 θ sin θ + sin8 θ cos θ =
sin(8 θ + θ)
= sin9 θ

4. Trig Identities By taking A = 60° and B = 30°,
Example 2
By taking A = 60° and B = 30°,
prove the identity for cos(A – B).
NB: cos(A – B) = cosAcosB + sinAsinB
LHS = cos(60 – 30 )°
= cos30°
= 3/2
RHS = cos60°cos30° + sin60°sin30°
= ( ½ X 3/2 ) + (3/2 X ½)
= 3/4 + 3/4
= 3/2
Hence LHS = RHS !!

5. Trig Identities Prove that sin15° = ¼(6 - 2) sin15° = sin(45 – 30)°
Example 3
Prove that sin15° = ¼(6 - 2)
sin15° = sin(45 – 30)°
= sin45°cos30° - cos45°sin30°
= (1/2 X 3/2 ) - (1/2 X ½)
= (3/2 /22)
= ( ) 22
X  2
= (6 - 2) 4
= ¼(6 - 2)

6. Trig Identities NAB type Question y x  
Example 4 y 41 x 3   4 40
Show that cos( - ) = 187/205
Triangle1
Triangle2
If missing side = x
If missing side = y
Then x2 = 412 – 402 = 81
Then y2 = = 25
So x = 9
So y = 5
sin = 9/41 and cos = 40/41
sin  = 3/5 and cos = 4/5

7. Trig Identities Remember this is a NAB type Question
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 )
= / /205
= /205
Remember this is a NAB type Question

8. Trig Identities A S T C xo 180+xo 360-xo 180-xo NAB type Question
Example 5
Solve sinxcos30 + cosxsin30 =
where 0o < x < 360o
ALWAYS work out Quad 1 first
By rule 1a sinxcos30 + cosxsin30 =
sin(x + 30) A S T C xo
180+xo
360-xo
180-xo
sin(x + 30) =
Quad 3 and Quad 4
sin = 75
Quad 3: angle = 180o + 75o
Quad 4: angle = 360o – 75o
x + 30o = 255o
x + 30o = 285o
x = 225o
x = 255o

9. Trig Identities A S T C θ  + θ 2  - θ  - θ sin2θ = 3/2
Example 6
Solve sin5 θ cos3 θ - cos5 θ sin3 θ = 3/2 where 0 < θ < 
By rule 1b. sin5θ cos3θ - cos5θ sin3θ =
sin(5θ - 3θ)
= sin2θ A S T C θ
 + θ
2  - θ
 - θ
sin2θ = 3/2
Quad 1 and Quad 2
Repeats every 
sin-1 3/2 = /3
Quad 1: angle = /3
Quad 2: angle =  - /3
In this example repeats lie out with limits
2 θ = /3
2 θ = 2/3
θ = /6
θ = /3

10. Trig Identities Find the value of x that minimises the expression
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. Trig Identities Paper 1 type questions
Example 8
Simplify sin(θ - /3) + cos(θ + /6) + cos(/2 - θ)
sin(θ - /3) + cos(θ + /6) + cos(/2 - θ)
= sin θ cos/3 – cos θ sin/3
+ cos θ cos/6 – sin θ sin/6
+ cos/2 cos θ + sin/2 sin θ
= 1/2 sin θ – 3/2cos θ + 3/2 cos θ – 1/2sin θ + 0 x cos θ + 1 X sin θ
= sin θ

12. (sinA + cosB)2 + (cosA - sinB)2 = 2(1 + sin(A - B))
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
= (sinAcosB - cosAsinB)
= sin(A – B)
= 2(1 + sin(A – B))
= RHS

13. Double Angle Formulae

14. Double Angle formulae Similarly: 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 C A S T 0o
180o
270o
90o

21. Trigonometric Equations

22. Trigonometric Equations-2 2 4 -4
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. Converting between degrees and radians:
Radian Measurements
Reminders
i) Radians
Converting between degrees and radians:

27. Degree Measurements Equilateral triangle: ii) Exact Values1 2
60o
30o
45o right-angled triangle: 1
45o

28. Radians / Degrees degrees 0o 30o 45o 60o 90o radians sin cos tan 1 1 11 1 1
Example:
What is the exact value of sin 240o ?

29. Sine Graph
Period = 360o
Amplitude = 1

30. Cosine Graph
Period = 360o
Amplitude = 1

31. Amplitude cannot be found for tan function
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 C A S T 0o
180o
270o
90o

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 C A S T 0o
180o
270o
90o
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. Solving Trigonometric Equations
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 C A S T 0o
180o
270o
90o
(2x – 60o ) = sin-1(1/2)
x 2
x 2

40. Solving Trigonometric Equations
Step 3: Solve the equation
1st quad
2nd quad
sin wave repeats every 360o
2x = 90o
210o
450o 570o
x = 45o
105o
225o
285o

41. Solving Trigonometric Equations
Graphical solution for

42. Solving Trigonometric Equations
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 C A S T 0o
180o
270o
90o
2 solutions
1st and 3rd quads
2 solutions
2nd and 4th quads

43. Solving Trigonometric Equations
Step 3: Solve the equation
1st quad
2nd quad
tan wave repeats every 180o
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 C A S T 0o
180o
270o
90o
Two solutions
One solution

46. Solving Trigonometric Equations
Step 3: Solve the equation
Two solutions
One solution
1stquad
2nd quad
90o
x = 19.5o
160.5o
Overall solution x = 19.5o , 90o and 160.5o

47. Solving Trigonometric Equations
Graphical solution for

48. Solving Trigonometric Equations
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 C A S T 0o
180o
270o
90o
Remember this !
Two solutions
One solution

49. Solving Trigonometric Equations
Step 3: Solve the equation
Two solutions
One solution
1stquad
3rd quad
180o
x = 53.1o
306.9o
Overall solution in radians x = 0.93 , π and 5.35

50. Solving Trigonometric Equations
Graphical solution for

51. Higher Maths Compound Angles Strategies Click to start
Higher Maths
Strategies
Compound Angles
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52. Compound Angles The following questions are on
Maths4Scotland Higher
The following questions are on
Compound Angles
Non-calculator questions will be indicated
You will need a pencil, paper, ruler and rubber.
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53. This presentation is split into two parts
Maths4Scotland Higher
This presentation is split into two parts
Using Compound angle formula for
Exact values
Solving equations
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54. Write down values for cos p and sin p
Maths4Scotland Higher
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. 8 4 p
Draw triangle
Pythagoras
Write down values for cos p and sin p
Expand sin (2p)
Expand cos (2p)
Hint
Use m = tan (2p)
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55. In triangle ABC show that the exact value of
Maths4Scotland Higher
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
Hint
Simplify
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56. cos x and sin x Maths4Scotland Higher
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
Hint
Simplify
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57. On the co-ordinate diagram shown, A is the point (6, 8) and
Maths4Scotland Higher
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). 10 13 6 8 5 12
Mark up triangles
Use Pythagoras
Write down values for
sin p, cos p, sin q, cos q
Expand sin (p + q)
Substitute values
Hint
Simplify
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58. A and B are acute angles such that and . Find the exact value of
Maths4Scotland Higher
A and B are acute angles such that and
Find the exact value of
a) b) c) 4 3 A 12 5 B 5 13
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)
Hint
Substitute
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59. Write down sin x and cos x
Maths4Scotland Higher
If x° is an acute angle such that
show that the exact value of 3 4 x 5
Draw triangle
Use Pythagoras
Hypotenuse is 5
Write down sin x and cos x
Expand sin (x + 30)
Substitute
Simplify
Hint
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Table of exact values

60. The diagram shows two right angled triangles
Maths4Scotland Higher
The diagram shows two right angled triangles
ABD 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 5
Use Pythagoras
Write down
sin x, cos x, sin y, cos y.
Expand cos (x + y)
Substitute
Hint
Simplify
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61. The framework of a child’s swing has dimensions
Maths4Scotland Higher
The framework of a child’s swing has dimensions
as shown in the diagram. Find the exact value of sin x°
Draw triangle
Draw in perpendicular
Use Pythagoras 3 4 x 2 h
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
Hint
Simplify
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62. cos a and sin a Maths4Scotland Higher Given that
find the exact value of 3 a
Draw triangle
Use Pythagoras
Write down values for
cos a and sin a
Expand sin 2a
Substitute values
Hint
Simplify
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63. Find algebraically the exact value of
Maths4Scotland Higher
Find algebraically the exact value of
Expand sin (q +120)
Expand cos (q +150)
Use table of exact values
Combine and substitute
Simplify
Hint
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64. cos q and sin q Maths4Scotland Higher If find the exact value of a) b)5 q 4 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
Hint
Find sin 4q
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65. Show that the exact value of
Maths4Scotland Higher
For acute angles P and Q
Show that the exact value of 12 13 P 5 3 Q 5 4
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
Hint
Simplify
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66. You have completed all 12 questions in this section
Maths4Scotland Higher
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67. Using Compound angle formula for
Maths4Scotland Higher
Using Compound angle formula for
Solving Equations
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68. Solve the equation for 0 ≤ x ≤  correct to 2 decimal places
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤  correct to 2 decimal places
Replace cos 2x with
Determine quadrants
Substitute A S C T
Simplify
Factorise
Hence
Discard
Hint
Find acute x
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69. Maths4Scotland Higher Equation Determine quadrants
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 A S C T
Rearrange
Check range
Find acute 2x
Hint
Deduce 2x
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70. a) Find expressions for: i) f(g(x)) ii) g(f(x))
Maths4Scotland Higher
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 A S C T
Determine
quadrants
Form equation
Replace sin 2x
Rearrange
Hint
Common factor
Hence
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71. Functions are defined on a suitable set of real numbers
Maths4Scotland Higher
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
1st expression
Simplifies to
2nd expression
Rearrange:
acute x
Simplify 1st expr. A S C T
Use exact values
Determine
quadrants
Similarly for 2nd expr.
Hint
Form Eqn.
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72. Solutions for where graphs cross
Maths4Scotland Higher
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 A S C T
Coords, P
Determine quadrants
for sin x
Hint
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73. Solutions are: x= 60°, 132°, 228° and 300°
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T A S C T
Factorise
Hence
Find acute x
Hint
Solutions are: x= 60°, 132°, 228° and 300°
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74. Solutions are: Maths4Scotland Higher Solve the equation for 0 ≤ x ≤ 2
Rearrange
Find acute x
Note range A S C T
Determine quadrants
Solutions are:
Hint
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75. Equal roots for cos q Maths4Scotland Higher
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
Replace cos 2q with
Try to solve:
Rearrange
Divide by 2
No solution
Hence there are no real solutions for q
Factorise
Equal roots for cos q
Deduction
Hint
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76. Solve algebraically, the equation sin 2x + sin x = 0, 0  x  360
Maths4Scotland Higher
Solve algebraically, the equation sin 2x + sin x = 0, 0  x  360
Replace sin 2x
Determine quadrants
for cos x
Common factor A S C T
Hence
Determine x
Hint
x = 0°, 120°, 240°, 360°
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77. Find the exact solutions of 4sin2 x = 1, 0  x  2
Maths4Scotland Higher
Find the exact solutions of 4sin2 x = 1, 0  x  2
Rearrange
Take square roots
Find acute x
Determine quadrants for sin x
+ and – from the square root requires all 4 quadrants A S C T
Hint
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78. Solutions are: x= 60°, 180° and 300°
Maths4Scotland Higher
Solve the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T
Factorise
Hence
Find acute x
Hint
Solutions are: x= 60°, 180° and 300°
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79. Solutions are: x= 60° and 300°
Maths4Scotland Higher
Solve algebraically, the equation for 0 ≤ x ≤ 360°
Replace cos 2x with
Determine quadrants
Substitute
Simplify A S C T
Factorise
Hence
Find acute x
Discard above
Hint
Solutions are: x= 60° and 300°
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80. You have completed all 12 questions in this presentation
Maths4Scotland Higher
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81. 30° 45° 60° sin cos tan 1 Table of exact values Maths4Scotland Higher
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82. You have completed all 12 questions in this presentation
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83. Are you on Target ! Update you log book
Make sure you complete and correct
ALL of the Trigonometry questions in the past paper booklet.