1. 2.0 TRIGONOMETRIC FUNCTIONS
TOPIC :
2.0 TRIGONOMETRIC FUNCTIONS
SUBTOPIC :
2.1 Introduction to Trigonometric Functions
LECTURE 1 OF 12

2. LEARNING OUTCOMES
a) recognize the graphs of y = sin x, y = cos x and y = tan x
b) recognize the graphs of y = a sin bx,
y = a cos bx and y = a tan bx.
c) understand the relationship between
graphs of sin x and cos x.

3. define trigonometric ratios
understand tan θ = ,
sin (90o – θ) = cos θ ,
cos (90o – θ) = sin θ and
tan (90o – θ) = cot θ
Use of some special angles
evaluate trigonometric functions for any angle

4. Definition Periodic Function
A function which graph consists of a basic pattern, which repeats at regular interval. The width of the basic pattern is the period of the function.

5. The value of sin , cos  and tan  will be repeated after one period,
the period of sin  and cos  is 2 rad and tan  is  rad.
So, the sine, cosine and tangent functions are known as periodic functions

6. Graph y = sin(x)

7. Graph y = sin (x) Period : 2 Range : [ -1, 1 ] Domain :
sin(-x) = - sin(x) x
-2 -  2
y=sin(x) 1 -1

8. Graph y = cos(x)

9. Graph of cos(x) is symmetry at y-axis X
Graph y = cos (x)
Period : 2
Range : [ -1, 1 ]
Domain :
Cos (-x) = cos (x)
Graph of cos(x) is symmetry at y-axis X
-2 -  2 y=
cos (x) 1 -1

10. Graph y = tan(x)

11. Graph y = tan (x) x y = tan (x) Period :  Range : Domain :
tan(-x) = - tan (x) x
y = tan (x) -1 1

12. GRAPH y = a sin bx y = a cos bx y = a tan bx.

13. y x

14. y x

15. y x

16. y x

17. Example 1 y x

18. Example 2 y x

19. Example 3 y x

20. y y x

21. Exercise
Sketch the graph of the following trigonometric functions: 1) 2) 3) 4)

22. The relationship between graphs of sin x & cos x

23. Graph one full period of sin x and cos xy
Graph one full period of sin x and cos x
y = -sin x 1 π 2π
π/2
3π/2 x –1
y = cos x
y = sin x
Shift to the right
sin (90o – θ) = cos θ

24. TRIGONOMETRY RATIO
For any acute angle θ, there are six trigonometry ratios, each of which is define by referring to a right angle triangle containing θ. y x r y
90o - θ θ x

25. x r θ y 90o - θ From the diagram y opposite sin θ = = r hypotenuse
adjacent x
cos θ = =
hypotenuse r
opposite y
tan θ = =
adjacent x

26. 1 r
cosec θ = =
sin θ y 1 r
sec θ = = x
cos θ 1 x
cot θ = =
tan θ y

27. Complimentary Angle r 90o - θ y θ x x sin (90o-θ) = = cos θ r y sin θ
cos (90o-θ) = = r x
cot θ
tan (90o-θ) = = y y
cot (90o-θ)
= tan θ r
90o - θ y θ x x

28. Trigonometric Ratios for Special Angle1 2
45o 1
60o 1

29. θo
θ rad
sin θ
cos θ
tan θ 0o
0 rad 1
30o
45o 1
60o
90o 1
undefined

30. Type of Angle
Acute angle
0o < θ < 90o
Right angle
θ = 90o
Obtuse Angle
90o < θ < 180o
Reflex angle
180o < θ < 360o

31. Positive and Negative Anglesy A
sin (-θ) = - sin θ S θ
cos (-θ) = cos θ x
tan (-θ) = - tan θ
- θ T C

32. Basic Angle (α) All positive Sine positive II θ I θ α =180o - θ α = θ
Cosine positive
Tangent positive

33. Example 1
If sin θ = and θ is acute angle, find
cos θ, tan θ, cosec θ, sec θ, and cot θ
Solution
Hence,
cot θ =
cos θ =
sec θ =
tan θ = 5 3
cosec θ = θ 4

34. Example 2
If tan θ = and θ is in the fourth quadrant
find the sec θ and cosec θ
Solution
Hence
sec θ = θ 1
cosec θ =

35. Example 3
Evaluate the following without using calculator
= - sin (210o -180o)
= - sin 30o =
(a) sin 210º

36. b) cos 120º
= - cos (180o – 120o)
= - cos 60o =
c) tan 240º
= tan (240o – 180o)
= tan 60o =

37. = - cos ( 225o – 180o)
= - cos 45o =
d) cos (- 225º)
e) cot (-300º)
= cot 60o =

38. CONCLUSION 2. The relationship between graphs of sin x & cos x
1. The graphs of
y = a sin bx y = a cos bx y = a tan bx.
2. The relationship between graphs of sin x & cos x
3. Define trigonometric ratios and understand
tan θ = ,
sin (90o – θ) = cos θ ,
cos (90o – θ) = sin θ and
tan (90o – θ) = cot θ
4. Use of some special angles
5. Evaluate trigonometric functions for any angle