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TOPIC :
2.0 TRIGONOMETRIC FUNCTIONS
SUBTOPIC :
2.1 Introduction to Trigonometric
Functions
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.
• define trigonometric ratios
sin 
• understand tan θ =
,
cos

o
sin (90 – 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
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.

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
Graph y = sin(x)
Graph y = sin (x)
 Period
: 2
 Range : [ -1, 1 ]
 Domain :  ,  
 sin(-x) = - sin(x)
x
-2
y=sin(x) 0
3
 
2
1
-


2
0
-1
0

2

3

2
2
0
1
0
-1
0
Graph y = cos(x)
y
1
x
0




-1



y
Graph y = cos (x)
1
 Period
: 2
 Range : [ -1, 1 ]
 Domain :  ,  
 Cos (-x) = cos (x)
 Graph of cos(x) is symmetry at y-axis
x
0







-1
X
y=
cos (x)
3



-2 2
- 
2
1
0
-1
0
0

2
3
 2

2
1
0
-1
0
1
Graph y = tan(x)
Graph y = tan (x)
Period : 
 Range : , 
n
 Domain : { \ 
; nis odd int egers}



2

tan(-x) = - tan (x)
x


2


4
y=
tan (x)

-1
0

4

2
0
1

GRAPH
y = a sin bx
y = a cos bx
y = a tan bx.
y
sin
a
sinbx
y
 asin
xx a
 00
y
a
1
0
1a
 2

2b2 2b
3
22b
24
2b
x
y  a sin bx a  0
y
a

2b
y  a sin bx a  0
3
2b
5
2b
x
0
a
2
2b
4
2b
y
cos
y
cos
x x
a
0
aacos
bx
a
0
y
a1
4

2b
3

2b
0
2



2b
2b
a1

22b
2 33

2b 22b
4
2

2b
x
y  a cos bx a  0
y
y  a cos bx
a0
a
4

2b
3

2b
2

2b


0
2b
a

2b
2
2b
3
2b
4
2b
x
y  3 cos 2 x
Example 1
a3 b2
y
3


3

4


2
2


0
4

4
3
3
4

x
y  2 cos 3x
Example 2
a  2 b  3
y
2
2

3


2


3


0
6
2


6
3

2
2
3
x
Example 3
3
y  3 sin x
2
3
a  3 b 
2
y
2b  3
3
0
3
5
3

3
2
3

4
3
x
y
y 
a tan
bx ax 0
y
tan
y
a
2

42b




4b4
a

44b
2 x
42b
y  a tan bx a  0
Exercise
Sketch the graph of the following
trigonometric functions:
1) y  sin 1 x
2
1
2) y   cos x
2
3) y   sin x
3
4) y  sin x
2
The relationship between
graphs of
sin x
&
cos x
Graph one full period of sin x
and cos x
y
y = -sin x
1
π/2
π
3π/2
2π
x
–1
y = cos x
Shift to the right

y = sin x
rad
o – θ) = cos θ
sin
(90
2
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
r
θ
90o - θ
y
x
x
From the diagram
r
θ
90o - θ
y
x
y
opposite
sin θ
=
hypotenuse
=
adjacent
cos θ
tan θ
=
hypotenuse
adjacent
r
x
=
r
y
opposite
=
x
=
x
cosec θ
1
=
sec θ
=
cot θ
=
sin θ
1
cos θ
1
tan θ
r
=
y
=
r
x
=
x
y
Complimentary Angle
x
=
• sin (90o-θ)
r
=
cos θ
=
sin θ
=
cot θ
y
• cos (90o-θ)
=
r
x
=
• tan (90o-θ)
y
y
• cot (90o-θ) = tan θ
r
90o - θ
y
θ
x
x
Trigonometric Ratios for Special Angle
45o
30o
2
1
2
3
45o
1
60o
1
θo
0o
30o
45o
60o
90o
θ rad
sin θ
cos θ
tan θ
0 rad
0
1
0

rad
6

rad
4

rad
3

rad
2
1
2
3
2
1
1
1
2
3
2
2
1
2
1
0
3
1
3
undefined
Type of Angle
Acute angle
Right angle
Obtuse Angle
Reflex angle
0o < θ < 90o
θ = 90o
90o < θ < 180o
180o < θ < 360o
Positive and Negative Angles
y
sin (-θ) = - sin θ
A
S
cos (-θ) = cos θ
θ
x
tan (-θ) = - tan θ
-θ
T
C
Basic Angle (α)
All positive
Sine positive
II
θ
I
θ
α =180o - θ
α=θ-
180o
α=θ
α
α
α
α =360o - θ
III
IV
θ
Tangent positive
Cosine positive
Example 1
If sin θ =
3
5
and θ is acute angle, find
cos θ, tan θ, cosec θ, sec θ, and cot θ
Solution
Hence,
cos θ =
tan θ =
5
3
θ
4
4
5
3
4
cosec θ =
cot θ =
sec θ =
3
5
4
3
5
4
Example 2
If tan θ = 
1
and θ is in the fourth quadrant
2
find the sec θ and cosec θ
Solution
Hence
2
sec θ =
θ
1
3
cosec θ =
1

cos 
1
 3
sin 
3
2
Example 3
Evaluate the following without using calculator
(a) sin 210º
= - sin (210o -180o)
= - sin 30o
=
1

2
b) cos 120º
= - cos (180o – 120o)
= - cos 60o
=
c) tan 240º

1
2
= tan (240o – 180o)
= tan 60o
=
3
d) cos (- 225º)
= - cos ( 225o – 180o)
= - cos 45o
= 
e) cot (-300º)
1
2
=
cot 60o
=
1
tan 60
=
1
3
CONCLUSION
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
sin 
tan θ =
,
cos 
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