1. Trigonometric Functions and Graphs
Higher Mathematics
Unit 1
Trigonometric Functions and Graphs

2. RADIANS So far we have always measured our angles in degrees.
There is another way to measure angles.
It is particularly important in applied mathematics.
Angles can be measured in
RADIANS

3. Radian Measure Length AB = radius
AOB subtends an arc equal to a radius

4. 1
2  radians = 360°
 radians = 180°

5.  radians = 180° So 1 radian =  So 1 radian ~ 57° ~ 180° =  radians2  2
Every 90° is radians

6.  2
90° 
180° 2 0° 3 2
270°

7. 180° =  radians Degrees to radians Change 60° to radians:3
60° = radians
(as 180°  ⅓ = 60°)

8.   180 Radians Degrees Convert 150° to radians
We can also convert as follows. 
180 
Degrees
Radians
Convert 150° to radians
(simplifying fraction: divide by 30)
150  
180
5  6
Radians

9. Change to Radians: 60° =  120° = Radians 210° = 2 315° = Radians
60  
180 2 3
Radians
120  
180 7 6
Radians
210  
180 7 4
Radians
315  
180

10.  radians = 180° Radians to Degrees: Change radians to degrees4
 radians = 180°  4 4
180°
radians =
= 45° 3 2
Change radians to degrees 3 2
radians =
= 270° 2
3 180°

11.   180 Degrees Radians Convert to degrees Radians
We can also convert as follows.
180  
Radians
Degrees 5 6
Radians
Convert to degrees
5   180
6  
5  180 6
150°

12. Change to degrees   180  45° Radians 2 Radians 120° 3 135°
4  
45° 2 3
Radians
2   180
3  
120° 3 4
Radians
3   180
4  
135°
5   180
3   5 3
Radians
300°

13.  radians = 180°  The angles in the following table must be known.
They are essential for non-calculator questions.
Degrees
360
180 90 60 45 30 2   2  3  4  6
Radians
remember as factors or multiples of 180°
 radians = 180°

14. Most angles in non-calculator work are multiples of those above
Use them to complete the table below
Degrees
120°
Use for Christines
Use smartboard document radians to angles doc
135°
210°
270°
315°
360° 2 3 5 6 5 4 4 3 5 3
Radians

15. Degrees Radians 120° 135° 150° 210° 225° 240° 270° 300° 315° 360° 2 3
Use for Christines only
135°
150°
210°
225°
240°
270°
300°
315°
360° 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4
11 6
Radians

16. Sketching Trig Graphs
For reinforcement and using the graphs go to folder graph transformations

17. Trig Graphs The maximum value for sin x is 1 when x = 90°
The minimum value for sin x is when x = 270°
sin x = 0 (i.e. cuts the x-axis) at:
x = 0°, x = 180°, x = 360

18. Trig Graphs  The maximum value for sin x is 1 when x =
The minimum value for sin x is when x =  2 3 2
sin x = 0 (i.e. cuts the x-axis) at:
x = x = x =  2

19. y = asinx

20. Trig Graphs
When sin x is multiplied by a number, that number gives the maximum and minimum value of the function.
Note the function still cuts the x-axis at: x = 0,  & 2

21. Trig Graphs The maximum value for cos x is 1 when x = 0° & 360°
The minimum value for cos x is when x = 180°
cos x = 0 (i.e. cuts the x-axis) at:
x = 90°, x = 270°

22. Trig Graphs The maximum value for cos x is 1 when x = 0 & 2
The minimum value for cos x is when x = 
cos x = 0 (i.e. cuts the x-axis) at:
x = x =  2 3 2

23. y = acosx

24. Trig Graphs
When cos x is multiplied by a number, that number gives the maximum and minimum value of the function.
Note the function still cuts the x-axis at: x =  2 3 2

25. Trig Graphs Using radians, sketch the following trig graphs: y = 5sinx
y = 1.5cosx
y = 2cosx
y = 100sinx
Use smartboard document graph transformations
When: 0 ≤ x ≤ 2p

30. y = -sinx

31. y = -sinx
The function y = -sinx is a reflection of y = sinx in the x - axis.

32. y = -cosx

33. y = -cosx
The function y = -cosx is a reflection of y = cosx in the x - axis.

34. y = sin nx

35. y = sin nx

36. Trig Graphs
PERIOD
PERIOD
When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p.
i.e. for y = sin nx: period of graph = 2p n

37. y = cos nx

38. Trig Graphs
PERIOD
PERIOD
When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p.
i.e. for y = cos nx: period of graph
For support of classwork use exercises on smartboard document trig transformations = 2p n

39. Trig Graphs Using radians, sketch the following trig graphs:
y = 5sin2x
y = 4cos2x
y = 6cos3x
y = 7sin½x
Use smartboard document graph transformations
When: 0 ≤ x ≤ 2p

44. Adding or subtracting from a Trig Function

48. Trig Graphs
When a number is added to a trig function the graph “slides” vertically up by that number.
When a number is subtracted from a trig function the graph “slides” vertically down by that number.
For support of classwork use exercises on smartboard document trig transformations

49. Trig Graphs Using radians, sketch the following trig graphs:
y = 3 + sin2x
y = cos3x - 4
y = 3sinx + 2
y = 2cos2x - 1
y = 2 - sinx
Use smartboard document graph transformations
When: 0 ≤ x ≤ 2p

55. Adding or subtracting from x

59. Trig Graphs
When a number is added to x the graph “slides” to the left by that number.
When a number is subtracted from x the graph “slides” to the right by that number by that number.
For support of classwork use exercises on smartboard document trig transformations

60. Example 1
Find the maximum turning point, for 0 ≤ x ≤ p, of the graph y = 5sin(x - p/3).
Consider the function y = 5sin x
Maximum value is 5
When x = p/2
For y = 5sin(x - p/3)
Max occurs at
p/2 + p/3 = 5p/6
Turning Point: (5p/6,5)

61. Example 2
Write down the equation of the drawn function and the period of the graph.
Write the function as y = asin bx + c

62. Example 2
y = asin bx + c
b = 3 (3 wavelengths in 2p)
Period of graph 2p  3
= 2p/3
Difference between max and min = 12
a = 12  2 = 6
y = 6sin 3x + c (graph then shifts up 2)
c = +2
y = 6sin 3x + 2

63. Ratios and Exact Values
Exact Values for 45° 1
Square 1
45° 1
45°

64. Ratios and Exact Values
Exact Values for 45° 1
45°
x² = 1² + 1²
x² = 2
x = √2 x √2

65. Ratios and Exact Values
Exact Values for 45° 1 √2
Sin 45° =
Cos 45° =
Tan 45° = 1
45° √2 1 √2 1

66. Ratios and Exact Values
Exact Values for p/4 1 √2
Sin p/4 =
Cos p/4 =
Tan p/4 = 1
p/4 √2 1 √2 1

67. Ratios and Exact Values
Exact Values for 30° & 60°
60°
30° 2 1
60°
30° 1 2
60° 2
Equilateral Triangle

68. Ratios and Exact Values
Exact Values for 30° & 60°
60°
30° 1 2
x² = 2² - 1²
x² = 3
x = √3 x √3

69. Ratios and Exact Values
Exact Values for 30°
60°
30° 1 2 1 2
Sin 30° =
Cos 30° =
Tan 30° = √3 2 √3 1 √3

70. Ratios and Exact Values
Exact Values for p/6
p/3
p/6 1 2 1 2
Sin p/6 =
Cos p/6 =
Tan p/6 = √3 2 √3 1 √3

71. Ratios and Exact Values
Exact Values for 60°
60°
30° 1 2
Sin 60° =
Cos 60° =
Tan 60° = √3 2 1 2 √3 √3

72. Ratios and Exact Values
Exact Values for p/3
p/3
p/6 1 2
Sin p/3 =
Cos p/3 =
Tan p/3 = √3 2 1 2 √3 √3

73. Angles Greater than 90° SIN positive ALL Positive 2nd Quadrant 1st
Sin A = (+)ve
Cos A = (-)ve
Tan A = (-)ve
1st
Quadrant
Sin A = (+)ve
Cos A = (+)ve
Tan A = (+)ve
180° 0°
3rd
Quadrant
TAN
positive
Sin A = (-)ve
Cos A = (-)ve
Tan A = (+)ve
4th
Quadrant
Sin A = (-)ve
Cos A = (+)ve
Tan A = (-)ve
COS
Positive
270°

74. Angles Greater than p/2 SIN positive ALL Positive TAN positive COS
TAN
positive
COS
Positive
3p/2

75. sin 3p/4 positive cos 7p/6 negative tan 7p/4 cos 5.4 radians TAN
ALL
Positive
COS
SIN 2p
sin 3p/4
cos 7p/6
tan 7p/4
cos 5.4 radians
positive
negative
For class support use smartboard document Four quadrants

76. 90°
SIN
ALL
(180 - x)° x°
180°
360°
(180 + x)°
(360 - x)°
COS
TAN
270°

77. Example 3
Solve 2sin x° = 1, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = sin x
2 sin x° = 1
sin x° = ½
For printing only see smatrboard document equations

78. Since sin x° is positive it is in the 1st and 2nd quadrants
Example 3
sin x° = ½
Since sin x° is positive it is in the 1st and 2nd quadrants
For printing only see smatrboard document equations

79. Example 3 sin x° = ½ √3 sin x° = ½ sin 30° = ½ x = 30° 30° 2 60° 1
For printing only see smatrboard document equations
60° 1

80. Example 3 sin x° = ½ sin 30° = ½ x = 30° x = 30° or x = 180° - 30°
For printing only see smatrboard document equations

81. Example 3
For printing only see smatrboard document equations

82. p/2
SIN
ALL
(p - )  p 2p
(p + )
(2p - )
COS
TAN
3p/2

83. Example 4
Solve √2cos  +1 = 0, 0 ≤  ≤ 2 and illustrate the solution in a sketch of y = cos 
√2cos  +1 = 0
√2cos  = -1
Cos  = √2 -1
For printing only see smatrboard document equations

84. Since cos  is negative it is in the 2nd and 3rd quadrants
Example 4
Cos  =
Since cos  is negative it is in the 2nd and 3rd quadrants √2 -1
For printing only see smatrboard document equations

85. Example 4 Cos  = √2 cos  = cos /4 =  = /4 1 p/4 1 -1 √2 1 √2 1 √2
For printing only see smatrboard document equations 1 √2 1

86. Example 4 Cos  = cos  = cos /4 = = /4 So  =  - /4 or  + /4√2 -1 √2 1
cos  =
cos /4 =
= /4
So  =  - /4 or  + /4
= 3/4 or 5/4 √2 1
For printing only see smatrboard document equations

87. Example 4
For printing only see smatrboard document equations

88. Consider if the equation was cos x = ½
Example 5
Solve cos 3x° = ½, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = cos 3x
Consider if the equation was cos x = ½
As cos x is positive it must be in the 1st and 4th quadrants.
For printing only see smatrboard document equations

89. Example 5 √3 cos x = ½ Cos 60° = ½ x = 60° or 360° - 60°
30° 2 √3
cos x = ½
Cos 60° = ½
x = 60° or 360° - 60°
x = 60° or 300°
For printing only see smatrboard document equations
60° 1

90. However the function we are using is cos 3x
Example 5
However the function we are using is cos 3x
Therefore if x = 60° or 300° for cos x = ½
3x = 60° or 3x = 300°: x = 20° or 100°
the graph repeats itself 3 times in 360°
with a wavelength of 120°
as the function has a wavelength of 120°
x = 20° or 100° or 140° or 220° or 260° or 340°
For printing only see smatrboard document equations

91. Example 5
For printing only see smatrboard document equations

92. and negative, x will be in all four quadrants
Example 6
Solve 2sin² x° = 1
sin² x = ½
sin x = √½
sin x = 
As sin x is positive
and negative, x will be in
all four quadrants √2 1
For printing only see smatrboard document equations

93. Example 6 sin x = sin 45° = x = 45° √2 1 45° 1 1 √2 1 √2
For printing only see smatrboard document equations
45° 1

94. Example 6 sin x = x = 45° or 180° - 45° or 180° + 45° or 360° - 45°√2 1
For printing only see smatrboard document equations

95. Factorise the equation Consider the equation as: 4x² + 11x + 6 =
Example 7
Solve 4sin²  + 11sin  + 6 = 0, correct to 2 decimal places, for 0 ≤  ≤ 2
Factorise the equation
Consider the equation as: 4x² + 11x + 6 =
(4x + 3)(x + 2) = 0
4sin²  + 11sin  + 6 = 0
(4sin + 3)(sin + 2) = 0
For printing only see smatrboard document equations

96. sin = or sin = -2 (no solution)
Example 7
4sin²  + 11sin  + 6 = 0
(4sin + 3)(sin + 2) = 0
4sin + 3 = 0 or sin+ 2 = 0
4sin = -3 or sin = -2
sin = or sin = -2 (no solution)
Therefore we have to solve sin = -0.75 4 -3
For printing only see smatrboard document equations

97. As sin is negative answer must be in 3rd and 4th quadrants
Example 7
sin = -0.75
As sin is negative answer must be in 3rd and 4th quadrants
sin = 0.75
 = sin-¹0.75 (radians)
 = 0.85 radians
For printing only see smatrboard document equations

98. Example 7  = 0.85 radians =  + 0.85 or  = 2 - 0.85
= 3.99 or 5.43 radians
For printing only see smatrboard document equations

99. Reminders:
sin² x° + cos² x° = 1
sin² x° = 1 - cos² x°
cos² x° = 1 - sin² x°

100. Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360
Example 8
Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360
(substitute cos² x° = 1 - sin²x° into the equation)
1 - sin² x° + sin x° = 1
1 - sin² x° + sin x° -1 = 0
sin x° - sin² x° = 0
sin x°(1 - sin x°) = 0
sin x° = 0 or 1 - sin x° = 0
sin x° = 1
x = 0°or 180° or 360° or x = 90°
For printing only see smatrboard document equations

101. Consider if the equation was sin x = 0.6 x = 36.87 or 180 - 36.87
Example 9
Solve sin (2x - 20)° = 0.6, correct to 1 decimal place, for 0 ≤ x ≤ 360
Consider if the equation was sin x = 0.6
x = or
x = 36.87° or °
For printing only see smatrboard document equations

102. The function we are considering is sin (2x - 20)
Example 9
x = 36.87° or °
The function we are considering is sin (2x - 20)
Therefore 2x - 20 = or 2x - 20 = ,
2x = or 2x =
x = 28.4° or x = 81.6°
For printing only see smatrboard document equations

103. The function repeats itself twice in 360°
Example 9
The function repeats itself twice in 360°
i.e. it has a wavelength of 180°
x = 28.4° or x = 81.6°
or x = ° or x = °
x = 28.4° or 81.6° or 208.4° or 261.6°
For printing only see smatrboard document equations

104. Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤  ≤ 
Example 10
Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤  ≤ 
Consider if the equation was 3cos x = 1
cos x = ⅓
 = 1.23 or 2 (remember to put calculator in radians)
 = 1.23 or
 = 1.23 or 5.05 radians
For printing only see smatrboard document equations

105. The function we are considering is cos(2 + /4)
Example 10
 = 1.23 or 5.05 radians
The function we are considering is cos(2 + /4)
2 + /4 = or 2 + /4 = 5.05
2 = or 2 =
2 = or 2 = 4.26
 = or  = 2.1 (to 1dp)
Do not need to add on a wave length of  as ≤  ≤ 
For printing only see smatrboard document equations