Algebra and Trigonometry

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 nGroups – 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.

COURSE LAYOUT

Weeks Weekly Lecture Topics1 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 andInfinite Series