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TRIGONOMETRIC RATIOS
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TRIGONOMETRIC
RATIOS

What is Trigonometry?
The word ‘trigonometry’ is derived from the ‘Greek’ words
i) tri
ii) gonia
iii) metron
‘tri’ means
three
‘gonia’
means an
angle or side
‘metron’
means
measure.
Hence, trigonometry means science of measuring ‘triangles’.

What is an Angle?
The amount of rotation of a ‘moving ray’ (terminating ray)
And it is denoted by
with reference to a ‘fixed ray’ (intial ray) is called an ‘angle’.
InItial ray
 or  or  etc.


If the rotation of the terminating ray is in anti clock wise direction,
Positive Angle
the angle is regarded as positive.
Initial ray


Negative Angle
If the rotation of the terminating ray is in clock wise direction,
the angle is regarded as negative
Initial ray
-

1) Trigonometry means science of _______________
1) angles
2) sides
3) triangles
4) polygons

2) If the rotation of the terminating ray is in anti clock wise
direction, the angle is regarded as _________________
1) Positive
2) Negative
3) Both
4) None of these

3) If the rotation of the terminating ray is in clock wise direction,
the angle is regarded as
1) Positive
2) Negative
3) Both
4) None of these

MEASUREMENT OF ANGLES
How to measure
an angle?
There are three systems to measure angles
i) The sexagesimal measurement.
ii) The centesimal measurement
iii) The circular measurement
British
system
French
system
Radian
system

i) The Sexagesimal measurement (British system)
In this system the unit of measurement of an angle is “degree”.
Degree
Each part
is called ‘a degree’, denoted by 10.
In this system one complete rotation is
divided into 360 equal parts.
𝟏
𝟑𝟔𝟎
𝒕𝒉
𝐩𝐚𝐫𝐭 = 𝟏𝟎
What is the
definition of a
degree?

Minute
A minute is further divided into
60 equal parts
and each part is called “one minute”, denoted as 1'
and each part is called “one second”, denoted as 1"
A degree is further divided into 60 equal parts
Second

Note
1)
1
360
of a complete rotation
2)
1
60
of a degree
3)
1
60
of a minute
4) One right angle
= 10( a degree)
= 1' ( a minute)
= 1" ( a second)
= 900

ii) The Centesimal Measurement (French system):
In this system the unit of measurement of an angle is
Grade
Each part
is called “a grade” denoted by 1g.
“grade”.
In this system one complete rotation is
divided onto 400 equal parts.
𝟏
𝟒𝟎𝟎
𝒕𝒉
𝐩𝐚𝐫𝐭 = 𝟏𝐠
What is the
definition of a
grade?

Minute
A minute is further divided into 100
equal parts
and each part is
called “a minute”, denoted as 1'
and each part is called “a second”, denoted as 1"
A degree is further divided into 100 equal parts
Second

1)
1
400
of a complete rotation
2)
1
100
of a grade
3)
1
100
of a minute
4) In this system one right angle
Note
= 1g( a grade)
= 1' ( a minute)
= 1"( a second)
= 100g

iii) The Circular measurement (Radian system):-
In this system the unit of measurement of an angle is
Radian
of the circle at its centre is called one radian, denoted by 1c.
“radian”.
The angle subtended by an arc of length equal to the radius
O
r
r
1c
r=l

1) Radian is a constant angle
2) One right angle
3) One radian
4) If no unit of measurement is indicated for an angle, it
will be understood that radian measure is implied
Note
=
𝟐
𝛑
right angle
=
𝛑
𝟐
radian

1) One minute in the centesimal system =____________ seconds
1) 60
2) 360
3) 400
4) 100

2) In the sexagesimal system one minute =__________ seconds
1) 360
2) 180
3) 90
4) 60

3) The centesimal system is also known as __________ system
1) British
2) French
3) Indian
4) American

Relation among the three systems
1) The formula connecting the three systems is as
follows,
2) One complete angle
3) One straight angle
4) One right angle
𝐃
𝟗𝟎𝟎
=
𝐆
𝟏𝟎𝟎𝐠
D=Degree
G=Grade
R=Radian
= 3600 = 400g = 2c
= 1800 =200g = c
= 900=100g =
Where D= degree, G= grade, R = radian
𝛑𝒄
𝟐
=
𝐑

𝟐
𝐜

5) 10
6) 1c
7) 10=
8) 1c=
𝟏𝟖𝟎𝟎
𝝅
= 0.01745c (approximately)
= 570 17' 45" (approximately)
𝛑𝒄
𝟏𝟖𝟎

1) 30°=__________ radians
𝟏)
𝛑
𝟒
𝟐)
𝛑
𝟑
𝟑)
𝛑
𝟐
𝟒)
𝛑
𝟔
Hint: We know that ,
𝐃
𝟗𝟎𝟎
=
𝐑
𝛑/𝟐
⇒
𝟑𝟎𝟎
𝟗𝟎𝟎
=
𝐑
𝛑/𝟐
⇒ 𝟑𝐑 =
𝛑
𝟐
⇒ 𝐑 =
𝛑
𝟔
∴ 𝟑𝟎𝟎 =
𝛑𝐜
𝟔

2) 45°=__________ gradians
1) 60 2) 50 3) 40 4) 30
Hint: We know that ,
𝐃
𝟗𝟎𝟎
=
𝐆
𝟏𝟎𝟎𝐠 ⇒
𝟒𝟓
𝟗𝟎
=
𝐆
𝟏𝟎𝟎
⇒ 𝟐𝐆 = 𝟏𝟎𝟎
⇒ 𝐆 = 𝟓𝟎
∴ 𝟒𝟓𝟎 = 𝟓𝟎𝐠

3) In the sexagesimal system straight angle =___________________
1) 360°
2) 180°
3) 90°
4) 60°

Trigonometric Ratios
The ratios of different pairs of sides of the right angled triangle are
called “trigonometric ratios” or “trigonometric functions”.

Take an angle of measure  in radian in the standard position.
Let P(x,y) be a point on the terminal side of the angle  such that
OP=r(>0)
O M
P(x,y)
x
y
r
X
Y
X’
Y’


Adjacent side (a)
Opposite
side (o)

A
B C
Opposite
side

sin  =
cos  =
tan  =
cosec  =
sec  =
cot  =
Adjacent
side
Hypotenuse
Adjacent side
Opposite side
Opposite side Hypotenuse
Adjacent side Hypotenuse
Opposite side Adjacent side
Consider the measure
of A to be 
For ,
Opposite side –
Adjacent side –
?
?
Side BC
Side AB
Let us consider all
three sides of right
angled ABC
With the help of
this, let us create 3
ratios
Can we create more
ratios ?
yes
3 more ratios can be created
What will be the
reciprocal of this
ratio ?
Hypotenuse
Opposite side
Hypotenuse
Opposite side
What will be the
reciprocal of this
ratio ?
Hypotenuse
Adjacent side
Hypotenuse
Adjacent side
What will be the
reciprocal of this
ratio?
Adjacent side
Opposite side
Adjacent side
Opposite side
Each of these ratios
are given a name.
They are…
So, in all 6 ratios can be
created
Let us see the ratios of
different pairs of sides

1) In right angled triangle tan=_______
𝟏)
𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐬𝐢𝐝𝐞
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞
𝟐)
𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐬𝐢𝐝𝐞
𝟑)
𝟒)

2) In right angled triangle cosec=___________
𝟏)
𝟐)
𝟑)
𝟒)

3) In right angled triangle cot=___________
𝟏)
𝐬𝐢𝐧𝛉
𝐜𝐨𝐬𝛉
𝟐)
𝐜𝐨𝐬𝛉
𝐬𝐢𝐧𝛉
𝟑)
𝟏
𝐜𝐨𝐬𝛉
𝟒)
𝟏
𝐬𝐢𝐧𝛉

 00,900,1800,2700,3600,……are called “quadrant angles”
Quadrant Angles and Allied Angles
 Two angles are said to be allied when their sum or difference is
either zero or a multiple of
π
2
 𝐓𝐡𝐞 𝐚𝐧𝐠𝐥𝐞𝐬 − 𝛉,
𝛑
𝟐
, 𝛑,
𝟑𝛑
𝟐
, 𝟐 etc., are called “allied angles”
to 

Note
2) For 900 , 2700 , we get co-ratios i.e.,
1) For 00 , 1800 , 3600 , we get same ratios
sin cos,
tan  cot,
sec cosec

Ratio Co-Ratio Reciprocal
sin cos cosec
cos sin sec
sec cosec cos
cosec sec sin
Reciprocal and Co-ratios of Trigonometric Ratios
tan cot cot
cot tan tan

1) Two angles are said to be allied when their sum or
difference is either ________________
1) zero
𝟐) 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐦𝐮𝐥𝐭𝐢𝐩𝐞 𝐨𝐟
𝛑
𝟐
3) Both 1 & 2
4) 𝐦𝐮𝐥𝐭𝐢𝐩𝐞 𝐨𝐟 𝛑

2) The co-ratio of cos is ___________
1) sin
2) sec
3) cot
4) cosec

3) Reciprocal of sin is………
1) sin
2) cos
3) cosec
4) cot

Signs of Trigonometric Functions
x’ x
y
y’
O
Q1
Q2
Q4
Q3
0°<  < 90°
360°+ 
90°
90°<  < 180°
180°- 
90°+
180°<  < 270°
180°+ 
270°
270°<  < 360°
270°+ 
360° or ( )
All
sin, cosec
cos, sec
tan, cot
Students
sin
&
Cosec
Take This
tan
&
Cot
Chart
cos
&
sec
Note:- With the
sentence “All Students
Take this Chart ” we
can remember the
signs of trigonometric
ratios
S
T C

Quadrant Q1 Q2 Q2 Q3 Q3 Q4 Q4 Q1 Q4
Allied angle
Tri. Ratios
90- 90+ 180- 180+ 270- 270+ 360- 360+ 
sin cos  cos  sin -sin -cos -cos -sin sin -sin
cos sin -sin  -cos -cos -sin sin cos cos cos
tan cot -cot -tan tan cot -cot -tan tan -tan
cot tan -tan -cot cot tan -tan -cot cot -cot
sec cosec -cosec -sec -sec -cosec cosec sec sec sec
cosec sec sec cosec -cosec -sec -sec -cosec cosec -cosec
Trigonometric Functions of Allied Angles

Trigonometric functions of 2n+ and 2n:
Trigonometric functions of 2n+ or 2n are same as 2+ or
2. (n Z)
Sin (2n+) = sin Sin (2n) =  sin
2n+  Q1 2n  Q4
Cos (2n+ ) = cos 
Tan (2n+ ) = tan 
Cot (2n+ ) = cot 
Sec (2n+ ) = sec 
Cosec (2n+ ) = cosec 
Tan (2n ) = tan
Cot (2n ) = cot 
Cosec (2n ) = cosec
Cos (2n ) = cos 
Sec (2n ) = sec 

Trigonometric Functions of n±
We can easily understand the trigonometric functions of n±
with the following diagram:
x’ x
y
y’
O
Q1
Q2
Q4
Q3
n
is
even
n+
n
n-
n+
n
is
odd

nN
Quadrant
Tr.Ratios
n is even n is odd
n
Q4
n+
Q1
n
Q2
n+
Q3
Sin sin sin sin sin
Cos cos cos cos cos
Tan tan tan tan tan
Cot cot cot cot
Sec sec sec sec sec
Cosec cosec cosec cosec cosec
cot

Values of Trigonometric functions from 0° to 90°
Angle
Ratios
00 300 450 600 900
0 1
Sin
1 0
Cos
0
Not
defined
1
Tan
Not
defined
1
Cot 0
1
Not
defined
2
Sec
Not
defined 2 1
Cosec
𝟏
𝟐
𝟏
𝟐
𝟑
𝟐
𝟑
𝟐
𝟏
𝟑
𝟑
𝟐
𝟑
𝟏
𝟐
𝟐
𝟐
𝟏
𝟐
𝟑
𝟏
𝟑
𝟐
𝟑

1) To which quadrant does 1800+ belong?
1) Q1
2) Q2
3) Q3
4) Q4

2) cos(270-)=
1) cos
2) -cos
3) sin
4) -sin

3) The value of tan600=
1)
𝟏
𝟑
2) 𝟑
3)
𝟏
𝟐
4) 1

The trignometric equations which are satisfied by all values of ‘’
in their domains are called
Trigonometric Identities
“trigonometric identities”.
sin2 + cos2 = 1
sec2 - tan2 = 1
cosec2 - cot2 = 1
They are

sin . cosec = 1
cos . sec = 1
tan . cot = 1
and

Domains and Ranges of Trigonometric Functions
Function Domain Range
Sinx R [-1,1]
Cosx R [-1,1]
Tanx R  {(2n+1)/2/nz} R
Cotx R  {n /nz} R
Secx R  {(2n+1)/2/nz} (, 1]  [1,)
Cosecx R  {n /nz} (, 1]  [1,)

1) Which trigonometric identity is true?
1) sin2 + cos2 = 1
2) sec2 - tan2 = 1
3) cosec2 - cot2 = 1
4) All the above

2) What is the formula for tan in terms of sec ?
𝟏) 𝟏 − 𝐬𝐞𝐜𝟐𝛉
𝟐) 𝟏 + 𝐬𝐞𝐜𝟐𝛉
𝟑) 𝐬𝐞𝐜𝟐𝛉 − 𝟏
4) 𝐬𝐞𝐜𝟐𝛉 + 𝟏
HINT
sec2-tan2=1
tan2= sec2 -1

3) What is the formula for cos in terms of sin ?
𝟏) 𝟏 − 𝐬𝐢𝐧𝟐𝛉
𝟐)
𝟏
𝐬𝐢𝐧𝛉
𝟑)
𝟏
𝟏 − 𝐬𝐢𝐧𝟐𝛉
𝟒)
𝟏 − 𝐬𝐢𝐧𝟐𝛉
𝐬𝐢𝐧𝛉

Thank you…

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