# Algebra and Trigonometry

By Dr Bijumon Ramalayathil   |   Mahatma Gandhi College, Iritty, Keezhur P.O., Kannur (Dist) , Kerala PIN: 670703
Learners enrolled: 1398
The course entitled “Algebra and Trigonometry” deals with the basic aspects of Algebra and Trigonometry.  The contents of this course are inevitable for many branches of sciences.  The students of Mathematics, Physics, Computer Science, Statistics, etc., are equally benefited with this course as a stepping stone to the broad areas of Algebra and Trigonometry.  The course aims to provide foundations in Algebra and Trigonometry.
The Course on Algebra and Trigonometry develops through the following topics:
Relations – Equivalence Relations – Partitions- –Functions – Congruent Modulo n
Groups – Properties of Groups - Subgroups-Cyclic Subgroups and Cyclic Groups- Permutations-Orbits and Cycles-Cosets and Theorem of Lagrange- Isomorphic Binary Structures-Group Homomorphisms-Rings and Fields-Integral Domains and Characteristic of a Ring.
Matrices-Adjoint and Inverse of Matrices – Rank of a Matrix and Elementary Transformations- Determination of Rank Using Transformations-Elementary Matrices-Fundamental Results Associated with Rank of Matrices- Inverse of a Matrix Using Transformations –Homogeneous System of Linear Equations-Method to Find Solution of Homogeneous System of Linear Equations-Solution of Non-Homogeneous System of Linear Equations – Characteristic Roots and Characteristic Vectors of a Matrix- Characteristic Space of a Matrix – Cayley-Hamilton Theorem.
Theory of Equations: The Fundamental Theorem of Algebra – The Occurrence of Imaginary Roots in Conjugate Pair - The Occurrence of Surd Roots in Conjugate Pair – Multiple Roots – Algorithm for Finding the Greatest Common Divisor of Two Polynomials – Relations Connecting the Roots and Coefficients of an Equation – Finding the Roots When they are connected by a relation – Determination of Values of Symmetrical Functions of the Roots – Sum of Powers of Roots of an Equation- Transformation of Equations – Reciprocal Equations – Nature of Roots – Newton’s Method of Approximation – Horner’s Process -  Depressing the Cubic Equation – Solving the Depressed Cubic – Cardano’s Solution of the Standard Cubic – Nature of the Roots of a Cubic – Quartic (or Biquadratic) Equation – Solution of Quartic Equations.
Trigonometry :  De-Moivre’s Theorem and Its Applications- Trigonometric Functions – Direct and Inverse Circular and Hyperbolic Functions – Logarithm of a Complex Quantity and Expansion of Trigonometric Functions –Summation of Series – Summation of Finite and Infinite Series.

The course is part of approved curriculum being transacted in College/University.
There will be 36 lectures of around half an hour duration each.
There will be an examination at the end of the course resulting awarding of grades.

Summary
 Course Status : Completed Course Type : Core Duration : 15 weeks Category : Credit Points : 4 Level : Undergraduate Start Date : 25 Jan 2021 End Date : 19 Apr 2021 Exam Date :

### Course layout

Weeks Weekly Lecture Topics
1 1 Relations
2 Equivalence Relations
3 Functions

2 4 Congruent Modulo n
5 Matrices
6 Adjoint and Inverse of Matrices

3 7 Rank of a matrix and Elementary Transformations
8 Determination of Rank using transformations
9 Elementary Matrices

4 10 Fundamental Results Rank and Inverse using transform
11 Solution of  Homogeneous System of Equations
12 Method to find solution of Homogeneous System of Equations

5 13 Non Homogeneous System of Equations
14 Characteristic Roots and Vectors
15 Characteristic Space

6 16 Cayley Hamilton Theorem
17 Theory of Equations  1
18 Theory of Equations 2

7 19 Theory of Equations  3
20 Theory of Equations 4
21 Theory of Equations 5

8 22 Groups
23 Isomorphic Binary Structures
24 Properties of Groups ad Subgroups

9 25 Cyclic subgroups and Cyclic Groups
26 Permutations
27 Orbits and Cycles

10 28 Cosets and Theorem of Lagrange
29 Homomorphisms
30 Rings and fields

11 31 Integral Domains and Characteristic of a ring
32 De Moivres Theorem
33 Direct and Inverse Circular Hyperbolic Functions

12 34 Logarithm of Complex Quantity Expansion of Trigonometric Functions
35 Summation of Series

13 36 Summation of Finite and Infinite Series

### Books and references

1. I.N. Herstein,  Topics in Algebra, Wiley Easten Ltd., New Delhi, 1975.
2. K. B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt. Ltd., New Delhi, 2000.
3. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul,  First Course in Linear Algebra,  Wiley Eastern, New Delhi, 1983.
4. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul,  Basic Abstract Algebra (2nd Edition),  Cambridge University Press, Indian Edition, 1997., New Delhi, 1983.
5. S.K. Jain, A. Gunawardena and P.B. Bhattacharya, Basic Linear Algebra with MATLAB, Key College Publishing (Springer-Verlag), 2001.
Peter V. O’ Niel, Advanced Engineering Mathematics, Cenage Learning, 2007.

### Instructor bio

Dr Bijumon Ramalayathil has 19 years of teaching experience both in UG and PG level.   He was awarded Ph.D. from the University of Calicut in 2010 for thesis titled On the Study of Frame Multiresolution Analysis in the Superspaces  under the guidance of Prof Dr. M. S. Balasubramani - University of Calicut.  His area of interest is Functional Analysis.  He got First Rank in Kerala Public Service Examination for the recruitment of Lecturers in 2004.  Presently he is Associate Professor and Head of Post Graduate Department of Mathematics, Mahatma Gandhi College, Iritty, Kerala.   He is the course coordinator of a couple of MOOC courses on SWAYAM platform.

He is the PI of the following Two MOOC (SWAYAM) Courses:
“Algebra and Trigonometry” for CEC through EMMRC University of Calicut (July-September 2018)
Differential Calculus for CEC through EMMRC University of Calicut (July-October 2019 More than 4800 learners joined for the course, July-October 2020 - 2956 learners joined for the course.) .  Also,
Subject Expert of more than 80 e-content modules (2011-2012 for CEC) on the Topics Basic Algebra, Abstract Algebra and  Vector Analysis and Geometry.
Subject Expert of DTH programme (for CEC) on Complex Analysis
Translation to Malayalam work of MOOC course on Algebra and Trigonometry  (2020)
Developed videos for SWAYAMPRABHA Channel
Developed study materials (BSc and MSc level) for many Universities in Kerala for their School of Distance Education.

### Course certificate

30% for in Course Assessment & 70% of End-term Proctored Exam.