1. Trigonometric Functions - Contents
Radian Measure
Area/Circumference/Length of Arc
Area of a Sector
Area of a minor segment
Small Angles
Trigonometric Results
Trigonometric Graphs
Graphical Solutions
Derivatives of Trigonometric Functions
Integrals of Trigonometric Functions
(Right Click Mouse  Pointer Options  Arrow Options  Visible)

2. Radian Measure
1 Unit
1 Unit 1c
One radian is the angle that a one unit arc makes with centre of a unit circle.
Proof
Circumference = 2πr
2π = 360o
π = 180o
= 2π (1)
π radians = 180o
= 2π
 The circumference is 2π

3. Radian Measure Conversions
Example 1
Convert into degrees. 5π 3
Example 2
Convert 45o into radians. 5π 3 =
5x180 3
π = 180o
= 300o
180o = π radians
1o = π/180 radians
Example 3
Convert 1.6c into degrees.
45o = π/180 x 45 radians
= 45π/180 radians
π radians = 180o
= π/4 radians
1 radians = 180/π deg
1.6 radians = 180/π x 1.6 deg
= 91o 40’

4. Area/Circumference/Length of Arc
Circumference = 2r
= D 
Area = r2 r
Length of Arc
l = r 
( is in radians)

5. Area/Circumference/Length of Arc
Circumference = 2r
= D
= 2 x 3
= 6 (exact)  4
≈ 18.8 cm
(Approximate)
Area = r2
3 cm
=  x 32
= 9 (exact)
Length of Arc
≈ 28.3 cm2
(Approximate)
l = r 
( is in radians)
= 3 x /4
= 3/4 (exact)
≈ 2.4 cm (Approximate)

6. Area of a Sector Area of Sector A = ½ r2  Proof Example
( is in radians)
Area of Sector
Area of Circle =
Angle 
Revolution
Example
Find the area of a sector with radius 3cm and angle /6. A
r2 =  2 A =
 r2 2
A = ½ r2 
A = ½ r2 
= ½ 32 x /6.
= 3/4 cm2.
≈ cm2.

7. Area of a Minor Segment Area of Sector A = ½ r2 ( - sin ) Exampleb a
Area = ½ab sin 
A = ½ r2 ( - sin )
( is in radians)
Example
Find the area of a minor segment formed with radius 3cm and angle /6.
Proof
Area of Minor Segment =
Area of
Sector -
Triangle
= ½ r2  - ½ r2 sin 
= ½ r2 ( - sin )
= ½ r2 ( - sin )
= ½ 32 x (/6 - sin /6)
= ½ 32 x (/6 - ½)
= (3/4 – 3/8)cm2
≈ cm2.

8. What happens to a/h as x 0.
Small Radian Angles
For small angles
Sin x ≈ x h
Tan x ≈ x x o
a=1
Cos x ≈ 1
Therefore
Cos x = a/h
lim Sin x = 1 x
x0
What happens to a/h as x 0.
lim Tan x = 1 x
x0
a/h  1/1 1

9. What happens to o/h as x 0.
Small Radian Angles
For small angles
Sin x ≈ x h
Tan x ≈ x x o
a=1
Cos x ≈ 1
Therefore
Sin x = o/h
lim Sin x = 1 x
x0
What happens to o/h as x 0.
lim Tan x = 1 x
x0
o/h x  0/1  0

10. What happens to o/a as x 0.
Small Radian Angles
For small angles
Sin x ≈ x h
Tan x ≈ x x o
a=1
Cos x ≈ 1
Therefore
Tan x = o/a
lim Sin x = 1 x
x0
What happens to o/a as x 0.
lim Tan x = 1 x
x0
o/a  0/1  0

11. Small Radian Angles Example: Evaluate: lim Sin 5x x lim Sin 5x=
lim 5(Sin 5x) 5x
x0
5 lim (Sin 5x) 5x
x0 =
= 5 x 1
= 5

12. Trigonometric Results
π/2
π - θ θ
2nd
1st S A
Note:
π=180o π 2π T C
3rd
4th
π + θ
2π - θ
3π/2

13. Trigonometric Results
Sin π/3 = √3/2
Sin π/6 = 1/2
Cos π/3 = 1/2
Cos π/6 = √3/2
60o
π/3
Tan π/3 = √3
Tan π/6 = 1/√3
π/6 2 2 √3
60o
π/3
60o
π/3 1 2
π/3 = 60o
π/6 = 30o

14. Trigonometric Results
Sin π/4 = 1/√2
Cos π/4 = 1/√2
π/4
45o
Tan π/4 = 1 √2 1
π/4
45o 1
π/4 = 45o

15. Trigonometric ResultsA S C T
Sin is -ve in the 3rd Quadrant.
Find the exact value of: 5π 4
5x180o 4
Sin
Sin =
= Sin 225o
= Sin (180o + 45o)
= -Sin 45o
= -1 √2

16. Trigonometric Results
Solve for 0 ≤ θ ≤ 2π A S C T
Evaluate Sin θ = 1/2
Sin is +ve in the 1st & 2nd Quadrants.
θ = 30o
30o 2 1
θ = π/6
θ = π - π/6
θ = 5π/6

17. Trigonometric Graphs
y = sin(x)
π/2
3π/2
Geogebra

18. Trigonometric Graphs
y = cos(x)
π/2
3π/2
Geogebra

19. Trigonometric Graphs
y = tan(x)
π/2
3π/2
Geogebra

20. Trigonometric Graphs
y = cosec(x)
π/2
3π/2
Geogebra

21. Trigonometric Graphs
y = sec(x)
π/2
3π/2
Geogebra

22. Trigonometric Graphs
y = cot(x)
π/2
3π/2
Geogebra

23. Trigonometric Graphs The a Sin x Family of Curves y = sin(x)
Geogebra

24. Trigonometric Graphs The Sin bx Family of Curves y = sin 3x y = sin 2x
Geogebra

25. Trigonometric Graphs The Sin x + c Family of Curves y = sin x
Geogebra

26. Graphical Solution Graphical solve y = cos x and y = x y = x y = cos x
One Solution
π/4
π/2

27. Derivative of Trigonometric Functions
Function of Function Rule
Derivative
Sin x d dx
[sin f(x)] = f’(x) cos f(x) d dx
(sin x) = cos x
Cos x d dx
(cos x) = sin x d dx
[cos f(x)] = -f’(x) sin f(x)
Tan x d dx
(tan x) = sec2 x d dx
[tan f(x)] = f’(x) sec2 f(x)

28. Derivative of Trigonometric Functions
Derivative Examples
Sin 4x d dx
(sin 4x) = 4 cos 4x d dx
[sin f(x)] = f’(x) cos f(x)
Tan (3x + 1) d dx
[tan (3x + 1)] = 3 sec2 (3x + 1) d dx
[tan f(x)] = f’(x) sec2 f(x)
Cos (x2 - 2x + 1) d dx
[cos (x2 - 2x + 1)] = -(2x - 2) sin (x2 -2x + 1)
= 2(1 - x) sin (x2 -2x + 1) d dx
[cos f(x)] = -f’(x) sin f(x)

29. Integrals of Trigonometric Functions
Function of Function Rule
Integral
Sin x
sin x dx = - cos x + C ∫ ∫
sin (ax +b) dx = - 1/a cos (ax + b) + C
Cos x
cos x dx = sin x + C ∫
cos (ax +b) dx = 1/a sin (ax + b) + C ∫
Tan x
sec2 x dx = tan x + c ∫
sec2 (ax +b) dx = 1/a tan (ax + b) + C ∫

30. Integral of Trigonometric Functions
Integration Examples ∫
sin (ax +b) dx = - 1/a cos (ax + b) + C ∫
sin (3x +2) dx = -1/3 cos (3x + 2) + C

31. Integral of Trigonometric Functions
Integration Examples
sin x dx = - cos x + C ∫ ∫
sin x dx = π 2
- cos x π 2
= - cos (-cos 0) π 2
= 0 – (-1)
= 1