1. Trigonometric functions
Mathematics content

2. Main ideas: circular measure of angles. angle, arc, sector
trig functions and their graphs
periodicity and amplitude
approximation to sin x, cos x, tan x when x is small
differentiating trig functions
primitives of trig functions
extension to functions of the form a sin (bx+c) etc

3. Radian Measure
A radian is defined as the angle made by taking the radius of a circle and ‘wrapping’ it around the circumference
radian demonstration
radian measure

4. So:
There are π radians in a half circle
And also 180° in a half circle
So π radians = 180°
So 1 radian = 180°/π = ° (approximately)
and also 1° = radians (approximately)
The circular measure of angles involves the measurement of angles in radians.

5. Express 90° in radians
180° = π radians
Find the equivalent of π radians in degrees
π radians = 180°
π radians = 180°
3 3
= 60°
1° = π radians
180
so 90°= 90 x π radians
= π radians 2
Evaluate cos using your calculator: (use RAD mode!)
cos =
= (to 3 dec pl)

6. NOTE: Exact value triangles make a reappearance in this topic!

7. Groves

9. Length of Circular Arc
In a circle of radius r units, an angle measuring θ radians stands on an arc of length l units. As l increases, so does θ, until the arc becomes the circumference (2πr) and the angle becomes a complete revolution (2π radians).
l = θ
2πr 2π
and l = rθ

10. Area of a Sector
The area of the shaded sector below is found as a fraction of the entire circle:
Area of sector = θ
πr π
A = ½ r2θ

11. e.g. Find the length of the arc in a circle of radius 6 cm, subtending an angle of 0.86 radians at the centre; and find the area of the corresponding sector
A = ½ r2θ
= ½ x 62 x 0.86
= cm2
l = rθ
= 6 x 0.86
= 5.16

14. Groves

16. Groves

19. Cambridge

22. Fitzpatrick
EXTENSION ONLY

25. Fitzpatrick
EXTENSION ONLY

26. Fitzpatrick
EXTENSION ONLY

27. Small Angles What happens when we use small angles, like 0.02 radians?
sin x = , tan x = cos x =
Thus sin x ≈ x ≈ tan x for small x and cos x ≈ 1
and it follows that:
lim x→0 sin x = 1 x

28. Cambridge

30. Graphs of Trigonometric Functions
We have already drawn graphs of trig functions...now we can draw them using radians:
y = sin x
y = cos x
y = tan x

31. And the reciprocal ratios:

32. Period and Amplitude
Trig curves repeat after a certain distance. This distance is called the period. The amplitude of the curve is half the distance between the maximum and minimum heights of the curve.

38. Groves

43. Cambridge

49. Differentiating Trigonometric Functions: sin x
Consider the gradient of the curve y = sin x
The gradient function is actually y = cos x

50. EXTENSION ONLY

51. Function of a Function Rule (Chain Rule)

53. Derivative of cos x We can similarly graph the gradient of cos x
The gradient function is y = -sin x

54. Derivative of tan x
The gradient function of y = tan x is harder to sketch:
The gradient function is actually y = sec2x

56. Groves

58. Cambridge

63. Cambridge

67. Primitives of Trigonometric Functions
Working backwards it is quite straightforward to find the primitives of trigonometric functions:

70. Groves

73. Cambridge

77. Cambridge

82. Integration of sin2x and cos2x
EXTENSION ONLY

84. Groves

86. Make sure you: create a summary of the topic
can work with angles, sectors and arcs
can differentiate and integrate trig functions and apply this to solving problems