PAPER NAME-ENGINEERING MATHEMATICS-I
PAPER CODE –BS 101
UNIT-III
Trigonometry
Angle
When a ray 𝑂𝐴 starting from its initial position 𝑂𝐴 rotates about its end point 𝑂
and takes the final position 𝑂𝐵, we say that angle 𝐴𝑂𝐵 (written as ∠𝐴𝑂𝐵 ) has
been formed.
The amount of rotation from the initial side to the terminal side is called the
measure of the angle.
Positive and Negative Angles
An angle formed by a rotating ray is said to be positive or negative depending on
whether it moves in an anti-clockwise or a clockwise direction, respectively.
Measurement of Angles
There are three system for measuring the angles, which are given below
1. Sexagesimal System (Degree Measure)
In this system, a right angle is divided into 90 equal parts, called the degrees. The
symbol 1∘ is used to denote one degree. Each degree is divided into 60 equal
parts, called the minutes and one minute is
divided into 60 equal parts, called the seconds. Symbols 1' and 1' are used to
denote one minute and one second, respectively.
i.e. 1 right angle = 90∘ , 1∘ = 60′ , 1′ = 60′′
2. Circular System (Radian Measure)
In this system, angle is measured in radian. A radian is the angle subtended at the
centre of a circle by an arc, whose length is equal to the radius of the circle. The
,number of radians in an angle subtended by an arc of circle at the centre is equal
arc
to radius .
3. Centesimal System (French System)
In this system, a right angle is divided into 100 equal parts, called the grades.
Each grade is subdivided into 100 min and each minute is divided into 100 s .
i.e. 1 right angle = 100 grades = 100𝑔 , 1𝑔 = 100′ , 1′ = 100′
Relation between Degree and Radian
(i) 𝜋radian= 180∘
180 ∘ 22
or 1 radian = = 5cos ∘ 1∘ 6′ 22′′ where, 𝜋 = = 3.14159
𝜋 7
𝜋
(ii) 1∘ = 180 rad = 0,01746rad
(iii) If 𝐷 is the number of degrees, 𝑅 is the number of radians and 𝐺 is the
𝐷 𝐺 2𝑅
number of grades in an angle 𝜃, then = 100 =
90 𝜋
Length of an Arc of a Circle
If in a circle of radius 𝑟, an arc of length 𝑙 subtend an angle 𝜃 radian at the centre,
then
𝑙 Length of arc
𝜃=𝑟= or 𝑙 = 𝑟𝜃
Radius
Trigonometric Ratios For acute Angle
Relation between different sides and angles of a right angled triangle are called
trigonometric ratios or T-ratios.
Trigonometric ratios can be represented as
, Perpendicular 𝐵𝐶
sin𝜃= = 𝐴𝐶 ,
Hypotenuse
Base 𝐴𝐵
cos𝜃= = 𝐴𝐶 ,
Hypotenuse
Perpendicular 𝐵𝐶
tan𝜃= = 𝐴𝐵 ,
Base
1
cosec𝜃= sin𝜃
1 cos𝜃 1
sec𝜃= cos𝜃 , cot𝜃 = = tan𝜃
sin𝜃
Trigonometric (or Circular) Functions
Let 𝑋 ′ 𝑂𝑋 and 𝑌𝑂𝑌 ′ be the coordinate axes. Taking 𝑂 as the centre and a unit
radius, draw a ciccle, cutting the coordinate axes at 𝐴, 𝐵, 𝐴′ and 𝐵′ , as shown in the
figure.
arc𝐴𝑃 𝜃 𝑙
∵ ∠𝐴𝑂𝑃 = radius𝑂𝑃 = 1 = 𝜃 ∘ , using 𝜃 = 𝑟
Now, six circular functions may be defined as
1
(i) cos𝜃 = 𝑥 (ii) sin𝜃 = 𝑦 (iii) sec𝜃 = 𝑥 , 𝑥 ≠ 0
1 𝑦 𝑥
(iv) cosec𝜃 = 𝑦 , 𝑦 ≠ 0 (v) tan𝜃 = 𝑥 , 𝑥 ≠ 0 (vi) cot𝜃 = 𝑦 , 𝑦 ≠ 0
Trigonometric Function of Some Standard Angles
Angle 0∘ 30∘ 45∘ 60∘ 90∘ 120∘ 135∘ 150∘ 180∘
1 1 √3 √3 1 1
sin 0 1 0
2 √2 2 2 √2 2
√3 1 1 1 1 √3
COS 1 0 − − − -1
2 √2 2 2 √2 2
PAPER CODE –BS 101
UNIT-III
Trigonometry
Angle
When a ray 𝑂𝐴 starting from its initial position 𝑂𝐴 rotates about its end point 𝑂
and takes the final position 𝑂𝐵, we say that angle 𝐴𝑂𝐵 (written as ∠𝐴𝑂𝐵 ) has
been formed.
The amount of rotation from the initial side to the terminal side is called the
measure of the angle.
Positive and Negative Angles
An angle formed by a rotating ray is said to be positive or negative depending on
whether it moves in an anti-clockwise or a clockwise direction, respectively.
Measurement of Angles
There are three system for measuring the angles, which are given below
1. Sexagesimal System (Degree Measure)
In this system, a right angle is divided into 90 equal parts, called the degrees. The
symbol 1∘ is used to denote one degree. Each degree is divided into 60 equal
parts, called the minutes and one minute is
divided into 60 equal parts, called the seconds. Symbols 1' and 1' are used to
denote one minute and one second, respectively.
i.e. 1 right angle = 90∘ , 1∘ = 60′ , 1′ = 60′′
2. Circular System (Radian Measure)
In this system, angle is measured in radian. A radian is the angle subtended at the
centre of a circle by an arc, whose length is equal to the radius of the circle. The
,number of radians in an angle subtended by an arc of circle at the centre is equal
arc
to radius .
3. Centesimal System (French System)
In this system, a right angle is divided into 100 equal parts, called the grades.
Each grade is subdivided into 100 min and each minute is divided into 100 s .
i.e. 1 right angle = 100 grades = 100𝑔 , 1𝑔 = 100′ , 1′ = 100′
Relation between Degree and Radian
(i) 𝜋radian= 180∘
180 ∘ 22
or 1 radian = = 5cos ∘ 1∘ 6′ 22′′ where, 𝜋 = = 3.14159
𝜋 7
𝜋
(ii) 1∘ = 180 rad = 0,01746rad
(iii) If 𝐷 is the number of degrees, 𝑅 is the number of radians and 𝐺 is the
𝐷 𝐺 2𝑅
number of grades in an angle 𝜃, then = 100 =
90 𝜋
Length of an Arc of a Circle
If in a circle of radius 𝑟, an arc of length 𝑙 subtend an angle 𝜃 radian at the centre,
then
𝑙 Length of arc
𝜃=𝑟= or 𝑙 = 𝑟𝜃
Radius
Trigonometric Ratios For acute Angle
Relation between different sides and angles of a right angled triangle are called
trigonometric ratios or T-ratios.
Trigonometric ratios can be represented as
, Perpendicular 𝐵𝐶
sin𝜃= = 𝐴𝐶 ,
Hypotenuse
Base 𝐴𝐵
cos𝜃= = 𝐴𝐶 ,
Hypotenuse
Perpendicular 𝐵𝐶
tan𝜃= = 𝐴𝐵 ,
Base
1
cosec𝜃= sin𝜃
1 cos𝜃 1
sec𝜃= cos𝜃 , cot𝜃 = = tan𝜃
sin𝜃
Trigonometric (or Circular) Functions
Let 𝑋 ′ 𝑂𝑋 and 𝑌𝑂𝑌 ′ be the coordinate axes. Taking 𝑂 as the centre and a unit
radius, draw a ciccle, cutting the coordinate axes at 𝐴, 𝐵, 𝐴′ and 𝐵′ , as shown in the
figure.
arc𝐴𝑃 𝜃 𝑙
∵ ∠𝐴𝑂𝑃 = radius𝑂𝑃 = 1 = 𝜃 ∘ , using 𝜃 = 𝑟
Now, six circular functions may be defined as
1
(i) cos𝜃 = 𝑥 (ii) sin𝜃 = 𝑦 (iii) sec𝜃 = 𝑥 , 𝑥 ≠ 0
1 𝑦 𝑥
(iv) cosec𝜃 = 𝑦 , 𝑦 ≠ 0 (v) tan𝜃 = 𝑥 , 𝑥 ≠ 0 (vi) cot𝜃 = 𝑦 , 𝑦 ≠ 0
Trigonometric Function of Some Standard Angles
Angle 0∘ 30∘ 45∘ 60∘ 90∘ 120∘ 135∘ 150∘ 180∘
1 1 √3 √3 1 1
sin 0 1 0
2 √2 2 2 √2 2
√3 1 1 1 1 √3
COS 1 0 − − − -1
2 √2 2 2 √2 2
