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Maths Microproject

Computer science and engineering (Shivaji University)

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Micro Project

SHIVAJIRAO JONDHALE POLYTECHNIC

INSTITUTE OF TECHNOLOGY AMBERNATH.

TRIGONOMETRY

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Program :- Informational
Technology

Program code :- FY-IT 1I

Course :- MATHEMATICS

Course code :-

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GROUP DETAILS
SR. NAME OF ROLL ENROLLMENT SEAT NO
GROUP
NO MEMBERS NO NO
1. YOGESH 516 2101470152
SHIRKE
2. VAIBHAVI 514 2101470145
PAWAR
3. MAYURI 509 2101470149
KAPSE
4. VARAD 513 2101470150
PATIL
5. SIDDHANT 507 2101470143
KADAM
6. ROHAN 506 2101470139
INGLE
7. VIVEK 517 2101470146
THOKAL
8. MOHIT 512 2101470142
PATIL
9. BHOOMIKA 511 2101470144
PATIL

NAME OF GUIDE : MR. RAVINDRA VARSALE

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Maharashtra State Board Of

Technical Education
Certificate.
This is to certify that Yogesh Anup Shirke Roll.no: 516 of 1st
semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.
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Maharashtra State Board Of

Technical Education
Certificate.
This is to certify that Mayuri Revansiddha Kapse Roll.no :
509 of 1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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Maharashtra State Board Of

Technical Education
Certificate.
This is to certify that Varad Ravindra Patil Roll.no : 513 of
1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.
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Maharashtra State Board Of

Technical Education Certificate.
This is to certify that Vaibhavi Satish Pawar Roll.no:514 of
1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale
Polytechnic, Ambernath (code:0147) has completed the
Micro Project Satisfactorily in Maths Academic Year
2021-2022 as prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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Maharashtra State Board Of

Technical Education
Certificate.
This is to certify that Siddhant Santosh Kadam Roll.no: 507
of 1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.
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Maharashtra State Board Of

Technical Education
Certificate.

This is to certify that Mohit Santosh Patil Roll.no: 512 of 1st

semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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Maharashtra State Board Of

Technical Education
Certificate.

This is to certify that Rohan Sudheer Ingle Roll.no: 506 of

1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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Maharashtra State Board Of

Technical Education
Certificate.

This is to certify that Bhoomika Sachin Patil Roll.no: 511 of

1st semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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Maharashtra State Board Of

Technical Education
Certificate.

This is to certify that Vivek Pravin Thokal Roll.no: 517 of 1st

semester of diploma in Information Technology
Engineering of Institute , Shivajirao Jondhale Polytechnic,
Ambernath (code:0147) has completed the Micro Project
Satisfactorily in Maths Academic Year 2021-2022 as
prescribed in the curriculum.

Place : Ambernath. Date : __________

Subject teacher. Head of the department.

Principal.

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:INDEX:
SR. TOPIC PG.
NO NO
1. INRODUCTION

2. FORMULAS

3. EXAMPLES

4. TEACHER’S EVALUTION SHEET

5. WEEKLY PROGRESS REPORT OF MICROPROJECT

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Introduction

Opposite side
Trigonometry is useful in our world. Lets see the
history of Trigonometry.
Trigonometry is derived from Greek word Trignon
(Three angles) and Metron (measure). The
Trigonometry is the branch of Mathematics B Adjacent side C
Which deals with triangles, particularly triangles in
a
plane where one
angle of triangle is 90°. Triangles on a sphere are
also studied in spherical
Trigonometry. Trigonometry specifically deals with
the relationship
Between the sides and the angles of Triangles that
is on the Trigonometric functions and with
calculations based on these functions.

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Hipparchus Bartholomaeus Piticus

Now the question is When the first Trigonometry was compiled?

The trigonometry is a field of mathematics first compiled in
2nd Century BCE by the Greek mathematician Hipparchus.
In 1595, the mathematician Bartholomaeus Piticus published an
influential work on Trigonometry in 1595, which may have coined the
Word Trigonometry.
In Indian astronomy the study of trigonometric
functions were flowered in Gupta Period,especially
due to Aryabhatta (6th century)

We know that there are six functions of an angle

commonly used in Trigonometry. Their names
and abbreviations are sine (sin), cosine (cos)
tangent (tan), cotangent (cot), secant (sec) and
cosecant (cosec). This all Trigonometric functions
are in relation to right triangle and the interesting
is The (Aryabhatiya) by Aryabhata. He is also
known as Father of Zero. Aryabhatta

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Now let us think we see the importance and

information about Trigonometry, But where
is
the use of Trigonometry?

 Trigonometry is used to set directions

such as north,south,east,west it tells you
what direction to wake with compass to
get on a straight direction. It is used in
Navigation to order to get pinpoint a
location.

 It is also used to find the distance of the

shore from a point in sea.

 It is used in Oceanography in calculating

the height of tides in Ocean.

 Trigonometry can be used to roof a

house, to make the Roof inclined (In case
of single individual in Bungalow) and the
Height of the roof in building etc.

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Basic formulas

1. sin² Θ + cos² = 1
2. 1 + tan²Θ = sec²Θ
3. 1 + cot²Θ = cosec²Θ
4. sin Θ = Opposite/hypotenuse
5. sec Θ = hypotenuse/adjacent sides
6. cosec Θ = hypotenuse/opposite sides
7. cos Θ = adjacent side/hypotenuse
8. tan Θ = opposite side/hypotenuse
9. cot Θ = adjacent side/opposite side
10. tan Θ = sin Θ/cos Θ
11. cot Θ = cos Θ/sin Θ
12. sec Θ = 1/cos Θ
13. cos Θ = 1/sec Θ
14. cosec Θ = 1/sin Θ
15. sin Θ = 1/cosec Θ

FORMULAE FOR (-Θ) :

1. sin (-Θ) = - sin Θ
2. cos (-Θ) = cos Θ
3. tan (-Θ) = - tan Θ
4. cot (-Θ) = - cot Θ
5. sec (-Θ) = sec Θ
6. cosec (-Θ) = - cosec Θ

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Signs Of Different Trigonometric

Ratios In Different Quadrants

Note: only cos (-Θ) and sec (-Θ) are positive (+) and remaining ratios
are negative (-).
y
90° = π/2

II (-,+) I

(+,+)

(π) 180° sin Θ and cos Θ } +ve ALL RATIOS ARE SAME
X
O
0° - 360 = 2π

tan Θ and cot Θ } +ve cos Θ and sec Θ } +ve

(+,-)
III IV

270° ( 3π/2)

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 NOTE: If Θ is any angle and

1. 0 < Θ < 90° or 0 < Θ < π/2 then Θ lies in 1st quadrant.
2. 90° < Θ < 180° or π/2 < Θ < π then Θ lies in II quadrant.
3. 180° < Θ < 270° or π < Θ < 3π/2 then Θ lies in III quadrant.
4. 270° < Θ < 360° or 3π/2 < Θ < 2π then Θ lies in IV quadrant.
5. π = 180°
STANDARD ANGLES:

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COMPOUND ANGLES :
An angle obtained by sum or the difference of two or more angles are
called compound angles.
e.g: A+B, A-B, x+y+z ….. Etc

ADDITION AND SUBTRACTION FORMULA:

1. sin (A+B) = sin A cos B + cos A Sin B
2. sin (A-B) = sin A cos B - cos A sin B
3. cos (A+B) = cos A cos B - sin A sin B
4. cos (A-B) + cos A cos B + sin A sin B
5. tan (A+B) = tan A + tan B
1 – tan A . tan B
6. tan (A-B) = tan A – tan B
1 + tan A . tan B

NOTE : If we put A – π/4 and B = Θ

7. tan ( π/4 + Θ ) = 1 + tan Θ (∵ tan π/4 = 1)
1 – tan Θ

8. tan ( π/4 - Θ ) = 1 – tan Θ

1 + tan Θ

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Some Useful results:

1. sin (A+B) . sin (A-B) = sin² (A+B) - sin² B

2. cos (A+B) . cos (A+B) = cos² A - sin² B

3. cos² A - cos² B = sin (A+B) . sin (A-B)

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Examples

Ex. Without using calculator, find the value of :-

a) Cos 15°
= cos (45°- 30°)
∵ cos (A-B) = cos A . cos B + sin A . sin B
= cos 45°. cos 30° + sin 45° . sin 30°
= 𝟏 𝐱 𝟑+ 𝟏 𝐱𝟏
𝟐 𝟐 𝟐 𝟐

𝟑 𝟏
= +
𝟐 𝟐 𝟐 𝟐

𝟑+𝟏
=
𝟐 𝟐

b) sin 75°
∵ sin (A+B) = sin A . cos B + cos A . sin B
= sin 45°. cos 30° + cos 45° . sin 30°
𝟏 𝟑 𝟏 𝟏
= 𝟐 𝒙 + 𝐱
𝟐 𝟐 𝟐

𝟑 𝟏
= +
𝟐 𝟐 𝟐 𝟐

𝟑+𝟏
=
𝟐 𝟐

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c) tan 105°

= tan (60°= 45°)

𝐭𝐚𝐧 𝟔𝟎°+𝐭𝐚𝐧 𝟒𝟓° 𝐭𝐚𝐧 𝑨=𝐭𝐚𝐧 𝑩
= {∵ 𝐭𝐚𝐧 𝑨 + 𝑩 } =
𝟏=𝐭𝐚𝐧 𝟔𝟎°−𝐭𝐚𝐧 𝟒𝟓° 𝟏−𝐭𝐚𝐧 𝑨 .𝐭𝐚𝐧 𝑩

𝟑+𝟏 𝟏+ 𝟑
= =
𝟏− 𝟑. 𝟏 𝟏− 𝟑

Ex.prove that
𝝅 𝝅 𝝅 𝝅
1) 𝐜𝐨𝐬⁡ − 𝑨 𝒄𝒐𝒔 + 𝑩 − 𝐬𝐢𝐧 − 𝑨 𝐬𝐢𝐧 + 𝑩 − 𝐬𝐢𝐧(𝑨 − 𝑩)
𝟔 𝟑 𝟔 𝟑

𝝅 𝝅
Put − 𝑨 = 𝒙 and + 𝑩 =𝒚
𝟔 𝟑

∴L.H.S = cos x . cos y – sin x . sin y

=cos (x+y)
𝝅 𝝅
cos − 𝑨+ + 𝑩
𝟔 𝟑

𝝅 𝝅
cos [ 30°-A +60°=B] ∵ = 𝟑𝟎°, = 𝟔𝟎°
𝟔 𝟑

cos[ 70° - (A-B)]

= sin (A-B)( ∵cos(90°-Θ) = sin Θ)

= R.H.S
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𝒔𝒊𝒏 (𝑨−𝑩) 𝒄𝒐𝒕 𝑨−𝒄𝒐𝒕 𝑩 𝐜𝐨𝐭 𝑩−𝐜𝐨𝐭 𝑨

2) = =
𝒄𝒐𝒔 (𝑨−𝑩) 𝟏+𝐜𝐨𝐭 𝑨 .𝒄𝒐𝒕𝑩 𝟏+𝐜𝐨𝐭 𝑨.𝐜𝐨𝐭 𝑩

𝒔𝒊𝒏 (𝑨−𝑩)
L.H.S =
𝒄𝒐𝒔 (𝑨−𝑩)

𝐬𝐢𝐧 𝑨 . 𝐜𝐨𝐬 𝑩−𝐜𝐨𝐬 𝑨 . 𝐬𝐢𝐧 𝑩

=
𝐜𝐨𝐬 𝑨 .𝐜𝐨𝐬 𝑩+𝐬𝐢𝐧 𝑨 . 𝐬𝐢𝐧 𝑩

Divide N and D by sin A . Sin B

𝐜𝐨𝐭 𝑩−𝐜𝐨𝐭 𝑨
=
𝐜𝐨𝐭 𝑨 .𝐜𝐨𝐭 𝑩
𝐜𝐨𝐭 + 𝑩−𝐜𝐨𝐭 𝑨
=
𝟏+𝐜𝐨𝐭 𝑨 .𝐜𝐨𝐭 𝑩

=R.H.S
𝒔𝒊𝒏 𝟐 𝜣 𝒄𝒐𝒔 𝟐 𝜣
3) –
𝒔𝒊𝒏 𝜣 𝒄𝒐𝒔 𝜣

𝒔𝒊𝒏 𝟐 𝜣.𝒄𝒐𝒔 𝜣 −𝒄𝒐𝒔 𝟐 𝜣 .𝒔𝒊𝒏 𝜣

=
𝒔𝒊𝒏 𝜣 .𝒄𝒐𝒔 𝜣
𝒔𝒊𝒏 ( 𝟐 𝜣− 𝜣)
= (∵ 𝐬𝐢𝐧 𝐀 − 𝐁 = 𝐬𝐢𝐧 𝐀 . 𝐜𝐨𝐬 𝐁 −
𝒔𝒊𝒏 𝜣 .𝒄𝒐𝒔 𝜣
𝐜𝐨𝐬 𝐀 . 𝐬𝐢𝐧 𝐁]
𝒔𝒊𝒏𝜣
=
𝒔𝒊𝒏 𝜣 .𝒄𝒐𝒔 𝜣

= sec 𝜣

= R.H.S
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4) 1 + cot Θ . cot 2 Θ = cot Θ . cosec 2 Θ

L.H.S = 1 + cot Θ . cot 2 Θ

𝐜𝐨𝐬 𝚯 .𝐜𝐨𝐬 𝟐 𝚯
=1+
𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯

𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯 +𝐜𝐨𝐬 𝚯 .𝐜𝐨𝐬 𝟐 𝚯

=
𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯
𝐜𝐨𝐬 𝚯 .𝐜𝐨𝐬 𝟐 𝚯 +𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯
=
𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯

𝒄𝒐𝒔 (𝟐 𝚯−𝟎) 𝒄𝒐𝒔

= =
𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯 𝒔𝒊𝒏 𝚯 . 𝐬𝐢𝐧 𝟐 𝚯
𝒄𝒐𝒔 𝚯 𝟏
= − = cosec 𝟐 𝚯
𝒔𝒊𝒏 𝚯 𝐬𝐢𝐧 𝟐 𝚯

= R.H.S

5) cos A . sin (B-C) + cos B sin (C-A) + cos C . sin (A-B) = 0

L.H.S = cos A . sin (B-C) = cos B . sin (C-A) + cos C . sin (A-B)

=cos A [sin B . cos C – cos B sin C] + cos B [ sin C cos A –cos C .

sin A ] +

= cos C [ sin A cos B – cos A sin B ]

= cos A sin B cos C – cos A cos B sin C = cos B sin C cos A –

cos B cos C sin A + cos C sin A cos B- cos C . cos A . sin B = 0

= R.H.S
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Pg. no : 37

SHIVAJIRAO JONDHALE POLITECHNIC

INSTITUTE OF TECHNOLOGY.
Academic year 2021-2022

Teacher Evaluation sheet for micro project of

physics.

Program Title : Information Technology.

Course Title & Course code: Basic Physics.(22102)

Semester : First.

Name of Students : 1) Vaibhavi.S.Pawar.(514).

2) Mayuri.R.Kapse.(509).
3) Bhoomika.S.Patil.(511).
4) Varad.R.Patil.(513).
5) Yogesh.A.Shirke. (516).
6) Mohit.S.Patil.(512).
7) Rohan.S.Ingle(506).
8) Siddhant.S.Kadam.(507).
9) Vivek.P.Thokal.(517).

Project Title : Comparison on Magnetism &

Electrostatics.

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Pg. no : 38

CO’s Addressed by the Micro-project : Use Basic

Knowledge of Magnetism and Electrostatics to
solve Engineering.

Major learning outcomes achieved by students by

doing the project :

a)Practical outcomes : I) Students Understood the

Concepts of Magnetism and Electrostatics.
II) Students Improve Their problems Solving Skill in
Group , discussion making skills and logical
reasoning.
III) Students have improves their management
skills . For example, time management, Teamwork,
etc.

b)Unit outcomes(In Cognitive domain) : Students

understood the concept of Magnetism and
Electrostatics. They can use knowledge of
Magnetism and Electrostatics in their real life.

c) Outcomes in affective domain :Students can

participate effectively in group work. Students also
learnt many things in Excel , PowerPoint , etc.Habit
of keeping record of event . Collect relevant
material , data from different sources.
Comments/Suggestions about team work /
Leadership / Inter-personal communication (If any)
Consistency in work , good co-ordination and
involvement in team.
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