Angle and Its Measurement
Trigonometry - 1
- Introduction of Trigonometry
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Fundamental Identities
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
- Trigonometric Functions of Sum and Difference of Angles
- Trigonometric Functions of Allied Angels
- Trigonometric Functions of Multiple Angles
- Trigonometric Functions of Double Angles
- Trigonometric Functions of Triple Angle
- Factorization Formulae
- Formulae for Conversion of Sum Or Difference into Product
- Formulae for Conversion of Product in to Sum Or Difference
- Trigonometric Functions of Angles of a Triangle
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Cramer’s Rule
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Introduction to Matrices
- Types of Matrices
- Algebra of Matrices
- Properties of Matrix Multiplication
- Properties of Transpose of a Matrix
Measures of Dispersion
Sequences and Series
Permutations and Combination
Methods of Induction and Binomial Theorem
Sets and Relations
A sequence `a_1, a_2, a_3,…, an,…` is called arithmetic sequence or arithmetic progression if `a_(n + 1) = a_n + d, n ∈ N`, where `a_1` is called the first term and the constant term d is called the common difference of the A.P.
The `n^(th)` term (general term) of the A.P. is` a^n = a + (n – 1) d`.
The sum to n term of A.P is `S_n= n/2[2a+(n-1)d]`
We can also write, `S_n = n/2[a+l]`
We can verify the following simple properties of an A.P. :
(1) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
(2) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
(4) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
(5) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.
Given two numbers a and b. We can insert a number A between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers a and b. Note that, in this case, we have
A – a = b – A, i.e., A =`(a+b)/2`
We may also interpret the A.M. between two numbers a and b as their average `(a+b)/2.`
For example, the A.M. of two numbers 4 and 16 is 10. We have, thus constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16.
The Arithmetic mean is `d = (b - a)/(n + 1)`