Subject:-Mathematics
Level:-10+2 and all entrance exams
Topic:-Trigonometric Functions and
Measurement of Angles
Angle : Let a revolving line starting from OA revolves
about its end point ‘O’ on a plane in clockwise or anti-
clockwise direction and occupy the position ‘OB’. It is
said to trace out ∠𝐴𝑂𝐵.
Here , OB is called the terminal side and OA is called
initial side. The fixed point ‘O’ is called vertex.
Measurement of Angles:
The amount of rotation from the initial side to the terminal side is called the measure of
the angle. If the rotation is in anti-clockwise sense, the angle measured is positive and if
the rotation is in clockwise sense, the angle measured is negative.
System of Measurement of Angles:
There are three major types of systems of measurement of angles which are as follows
1. Sexagesimal system (degree measure)- In this system a right angle is divided into
90 equal parts, called degrees. Each degree is divided into 60 equal parts, called
minutes and each minute is further divided into 60 equal parts, called seconds.
Thus, 1 right angle=90 degrees=(900 ), 10 = 60 min = (60′ ), 1′ = 60s = (60′′ )
2. Centesimal system- In this system a right angle is divided into 100 equal parts,
called grades. Each grade is divided into 100 centesimal minutes and each minute is
further divided into 100 centesimal seconds.
Thus, 1 right angle = 100 grades = (100g )
, 1 grade = 100min = (100′ )
1min=100s=(100′′)
3. Circular system- In this system the unit of measurement is radian as defined below
i. One radian, written as 1𝑐 , is the measure of an angle subtended at the centre
of a circle by an arc of length equal to radius of the circle.
ii. The number of radians in an angle subtended by an arc of a circle at the centre
𝑎𝑟𝑐
is equal to .
𝑟𝑎𝑑𝑖𝑢𝑠
Relation between Degrees, Grades and Radians
𝐷𝑒𝑔𝑟𝑒𝑒 𝑔𝑟𝑎𝑑𝑒
The relation between the three systems of measurement of an angle is = =
90 100
2 𝑟𝑎𝑑𝑖𝑎𝑛
𝜋
Thus,
180
i. To convert radians into degrees multiply by ( ).
𝜋
𝜋
ii. To convert degrees into radians multiply by ( ).
180
Example 1. Find the angle between the minute hand of a clock and the hour hand when the
time is 7:20am?
Ans: We know that, the hour hand completes one rotation in 12h while the minute hand
completes one rotation in 60min.
∴ Angle traced by the hour hand in 12 h = 3600
20 1 22
Now the time is 7:20 am i.e. 7h 20 min= 7 + ( ) h = 7 + = h
60 3 3
⇨ Angle traced by the hour hand in 7 h 20 min.
22 360 22
i.e.
3
h=(
12
×
3
) = 2200
Also, the angle traced by the minute hand in 60min =3600
360 0
The angle traced by the minute hand in 20 min= ( × 20) = 1200
60
∴ Required angle between two hands = 2200 − 1200 = 1000
Level:-10+2 and all entrance exams
Topic:-Trigonometric Functions and
Measurement of Angles
Angle : Let a revolving line starting from OA revolves
about its end point ‘O’ on a plane in clockwise or anti-
clockwise direction and occupy the position ‘OB’. It is
said to trace out ∠𝐴𝑂𝐵.
Here , OB is called the terminal side and OA is called
initial side. The fixed point ‘O’ is called vertex.
Measurement of Angles:
The amount of rotation from the initial side to the terminal side is called the measure of
the angle. If the rotation is in anti-clockwise sense, the angle measured is positive and if
the rotation is in clockwise sense, the angle measured is negative.
System of Measurement of Angles:
There are three major types of systems of measurement of angles which are as follows
1. Sexagesimal system (degree measure)- In this system a right angle is divided into
90 equal parts, called degrees. Each degree is divided into 60 equal parts, called
minutes and each minute is further divided into 60 equal parts, called seconds.
Thus, 1 right angle=90 degrees=(900 ), 10 = 60 min = (60′ ), 1′ = 60s = (60′′ )
2. Centesimal system- In this system a right angle is divided into 100 equal parts,
called grades. Each grade is divided into 100 centesimal minutes and each minute is
further divided into 100 centesimal seconds.
Thus, 1 right angle = 100 grades = (100g )
, 1 grade = 100min = (100′ )
1min=100s=(100′′)
3. Circular system- In this system the unit of measurement is radian as defined below
i. One radian, written as 1𝑐 , is the measure of an angle subtended at the centre
of a circle by an arc of length equal to radius of the circle.
ii. The number of radians in an angle subtended by an arc of a circle at the centre
𝑎𝑟𝑐
is equal to .
𝑟𝑎𝑑𝑖𝑢𝑠
Relation between Degrees, Grades and Radians
𝐷𝑒𝑔𝑟𝑒𝑒 𝑔𝑟𝑎𝑑𝑒
The relation between the three systems of measurement of an angle is = =
90 100
2 𝑟𝑎𝑑𝑖𝑎𝑛
𝜋
Thus,
180
i. To convert radians into degrees multiply by ( ).
𝜋
𝜋
ii. To convert degrees into radians multiply by ( ).
180
Example 1. Find the angle between the minute hand of a clock and the hour hand when the
time is 7:20am?
Ans: We know that, the hour hand completes one rotation in 12h while the minute hand
completes one rotation in 60min.
∴ Angle traced by the hour hand in 12 h = 3600
20 1 22
Now the time is 7:20 am i.e. 7h 20 min= 7 + ( ) h = 7 + = h
60 3 3
⇨ Angle traced by the hour hand in 7 h 20 min.
22 360 22
i.e.
3
h=(
12
×
3
) = 2200
Also, the angle traced by the minute hand in 60min =3600
360 0
The angle traced by the minute hand in 20 min= ( × 20) = 1200
60
∴ Required angle between two hands = 2200 − 1200 = 1000
