Welcome To Basics In Maths

Ts Inter Maths IA Concept

maths IA concept feature image

This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate first year Math students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E  first year maths 1A syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.


1.Functions

Set: A collection of well-defined objects is called a set.

Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

Cartesian product: Let A and B are two non-empty sets. The Cartesian product of A and B is denoted by A × B and is defined as a set of all ordered pairs (a, b) where a ϵ A and b ϵB

                           cartesion product              

Relation: Let A and B are two non-empty sets the relation R from A to B is a subset of A×B.

⇒ R: A→B is a relation if  R⊂ A × B

Function:

A relation f: A → B is said to be a function if ∀ aϵ A there exists a unique element b such that (a, b) ϵ f.                                            (Or) 

A relation f: A → B is said to be a function if 

(i) x ϵ A ⇒ f(x) ϵ B 

(ii)  x1 , x2 ϵ A , x1 = x2 in A  ⇒ f(x1) = f(x2) in B.

 Note:   If A, B are two finite sets then the no. of   functions that can be defined from A to B is  n(B)n(A)

VARIOUS TYPES OF FUNCTIONS

One– one Function (Injective):- A function f: A→ B is said to be a one-one function or injective if different elements in A have different images in B.

(Or)

A function f: A→ B is said to be one-one function if f(x1) = f(x2) in B ⇒ x1 = x2 in A.

Note: No. of one-one functions that can be defined from A into B is n(B) p n(A)   if  n(A) ≤ n(B)

On to Function (Surjection): – A function f: A→ B is said to be onto function or surjection if for each yϵ B ∃ x ϵ A such that f(x) =y

onto function

Note: if n(A) = m and n(B) = 2 then no. of onto functions = 2m – 2

 Bijection: – A function f: A→ B is said to be Bijection if it is both ‘one-one and ‘onto’.

bijection

Constant function:  A function f: A→ B is said to be constant function if f(x) = k ∀ xϵA

Identity function:  Let A be a non-empty set, then the function defined by I: A → A, I(x)=x is called identity function on A.

Equal function:  Two functions f and g are said to be equal if

(i)   They have same domain (D)

(ii)  f(x) = g(x) ∀ xϵ D

Even function:  A function f: A→ B is said to be even function if f (- x) = f(x) ∀ xϵ A

Odd function:   A function f: A→ B is said to be odd function if f (- x) = – f(x) ∀ xϵ A

Composite function:  If f: A→B, g: B→C are two functions then the composite relation is a function from A to C.

composite function

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).

Step function:  A number x = I + F

 I → integral part    = [x]

F → fractional part = {x}

∴ x = [x] + {x}

step functionIf y = [x] then domain = R and

Range = Z                                                 

0 ≤ x ≤ 1, [x] = 0

1≤ x ≤ 2, [x] = 1

-1 ≤ x ≤ 0, [x] = -1

If k is any integer [ x + k] = k + [x]

The value of [x] is lies in x – 1 < [x] ≤ 1.

Inverse function: If f: A → B is bijection then f -1  is exists

f-1: B → A is an inverse function of f.

domain and range

in equations

SOME IMPORTANT POINTS

of subsets of a set of n elements is 2n

of proper subsets of a set of n elements is 2n – 1

Let A and B are two non-empty finite sets and f: A → B is a function. This function will

One-one if n(A) ≤ n(B)

On to if n(A) ≥ n(B)

Bijection   if n(A) = n(B).


2. MATHEMATICAL INDUCTION


3. MATRICES

Matrix: An ordered rectangular array of elements is called a matrix

  • Matrices are generally enclosed by brackets like
  • Matrices are denoted by capital letters A, B, C and so on
  • Elements in a matrix are real or complex numbers; real or complex real-valued functions.

Oder of Matrix: A matrix having rows and ‘n’ columns is said to be of order m x n. Read as m by n.

matrix

Types of Matrices

 Square Matrix: A matrix in which the no. of rows is equal to the no. of columns is called a square matrix.

square matrix
 Principal diagonal ( diagonal)  Matrix: If A  = [aij] is a square matrix of order ‘n’ the elements  a11 , a22 , a33 , ………. ann is said to constitute its principal diagonal.

principle diagonal matrix

Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).

Ex:-

trace of matrix

Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

diagonal of matrix

Scalar Matrix: If each non-diagonal elements of a square matrix are ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.

scallar matrix

Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called a unit matrix.

identity matrix

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.

null matrix

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.

row and column matrices

Triangular matrices:

A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j

A square matrix A = [aij] is said to be lower triangular matrix aij = 0  ∀ i < j

triangular matrices

Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.

equality of matrices

Product of Matrices:

 Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp  where

product of matrices

Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT.  

transpose of matrix

Note: (i) (AI)I = A (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT

Symmetric Matrix: A square matrix A is said to be symmetric if AT =A

If A is a symmetric matrix, then A + AT is symmetric.

Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A

If A is a skew-symmetric matrix, then A – AT is skew-symmetric

Minor of an element: Consider a square matrix   

minor of an elemen

the minor an element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.

minor of an element example

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as it’s minor multiplied by (- 1 ) i+j .

Properties of determinants:

  • If each element of a row (column) of a square matrix is zero, then the determinant of that matrix is zero.

det-1

  • If A is a square matrix of order 3 and k is scalar then.
  • If two rows (columns) of a square matrix are identical (same), then Det. Of that matrix is zero.

det-2

  • If each element in a row (column) of a square matrix is the sum of two numbers then its determinant can be expressed as the sum of the determinants.

         det-3

  • If each element of a square matrix are polynomials in x and its determinant is zero when x = a, then (x-a) is a factor of that matrix.
  • For any square matrix A  Det(A) =  Det (AI).
  • Det(AB) = Det(A) . Det(B).
  • For any positive integer n Det(An) = (DetA)n.

Singular and non-singular matrices: A Square matrix is said to be singular if its determinant is zero, otherwise it is said to be the non-singular matrix.

singular and non-singular matrices

Ad joint of a matrix: The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.

adjoint of matrix 2

 Invertible matrix: Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is the unit matrix of the same order as A and B.

invertible matrix

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].

augmented matrix

Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.

sub matrix

Let A be a non-zero matrix. The rank of A is defined as the maximum of the order of the non-singular submatrices of A.

  • Note: If A is a non-zero matrix of order 3 then the rank of A is:
  • 1, if every 2×2 submatrix is singular
  • 2, if A is singular and at least one of its 2×2 sub-matrices is non-singular

   (iii)  3, if A is non – singular.

Consistent and Inconsistent: The system of linear equations is consistent if it has a solution, in-consistent if it has no solution.

  • Note: The system of three equations in three unknowns AX = D has
  • A unique solution if rank(A) = rank ([AD]) = 3
  • Infinitely many solutions if rank (A) = ([AD]) < 3
  • No solution if rank (A) ≠ rank ([AD])

Solutions of a homogeneous system of linear equations:

hogenious equations

The system of equations AX = 0 has

  • The trivial solution only if rank(A) = 3
  • An infinite no. of solutions if rank(A) < 3

4.ADDITION OF VECTORS

Directed line: If A and B are two distinct points in the space, the ordered pair (A, B) denoted by AB is called a directed line segment with initial point A and terminal point B.

directed line

⇒ A directed line passes through three characteristics: (i) length (ii) support (iii) direction

Scalar: A quantity having magnitude only is called a scalar. We identify real numbers as a scalar.

Ex: – mass, length, temperature, etc.

Vector: A quantity having length and direction is called a vector.

Ex: – velocity, acceleration, force, etc.

⇒ If TS inter addition of vectors 4 is a vector then its length is denoted by TS inter addition of vectors 28

Position of vector: If P (x, y, z) is any point in the space, then TS inter addition of vectors 1 is called the position vector of the point P with respect to origin (O). This is denoted by TS inter addition of vectors 2

Like and unlike vectors:  If two vectors are parallel and having the same direction then they are called like vectors.

like vectors

 

If two vectors are parallel and having opposite direction then they are called, unlike vectors.


un like vectors
Coplanar vectors:
Vectors whose supports are in the same plane or parallel to the same plane are called coplanar vectors.

VECTOR ADDITION                                       

Triangle law: If TS inter 1A product of vectors 2 are two vectors, there exist three points A, B, and C in a space such that   defined by TS inter addition of vectors 7

triangle law

Parallelogram law: If two vectorsTS inter 1A vector a and TS inter addition of vectors 5 represented by two adjacent sides of a parallelogram in magnitude and direction then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.

parallelogram law 2                                                                    parallelogram law

Scalar multiplication: LetTS inter 1A vector a be a vector and λ be a scalar then we define vector λTS inter 1A vector a  to be the vectorTS inter addition of vectors 29 if eitherTS inter 1A vector a is zero vector or λ is the scalar zero; otherwise λTS inter 1A vector a is the vector in the direction of TS inter 1A vector awith the magnitude TS inter addition of vectors 9if λ>0 and λTS inter 1A vector a  = (−λ)(−TS inter 1A vector a ) if λ<0.

add. vectors notes

The angle between two non-zero vectors:   LetTS inter 1A product of vectors 2 be two non-zero vectors, let TS inter addition of vectors 10  then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectorsTS inter 1A vector a and TS inter addition of vectors 5, it is denoted by (TS inter 1A product of vectors 2 ).

TS inter addition of vectors 12

Section formula: LetTS inter 1A product of vectors 2 be two position vectors of the points A and B with respect to the origin if a point P divides the line segment AB in the ratio m:n then

section formula

Linear combination of vectors:  let TS inter addition of vectors 13 be vectors x1, x2, x3…. xn be scalars, then the vectorTS inter addition of vectors 14 is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectorsTS inter addition of vectors 15 If r is any vector then there exist a unique triad (x, y, z) of scalars such that TS inter addition of vectors 16 . These scalars x, y, z are called the components of TS inter addition of vectors 2with respect to the ordered triad   (a, b, c).

  • i, j, k are unit vectors along the X, Y and Z axes respectively and P(x, y, z) is any point in the space thenTS inter addition of vectors 1 = r = x i + y j +z k   andTS inter addition of vectors 17

Regular polygon: A polygon is said to be regular if all the sides, as well as all the interior angles, are equal.

  • If a polygon has sides then the no. of diagonals of a polygon is TS inter addition of vectors 18 
  • The unit vector bisecting the angle between  is  TS inter addition of vectors 19

Vector equation of a line and plane

⇒The vector equation of the line passing through point A (TS inter 1A vector a) and ∥el to the vector TS inter addition of vectors 5 is

vector equation of a line

Proof:-

vector equation opf a line 2

 Then AP,  are collinear vector proof: let P (TS inter addition of vectors 2 ) be any point on the line a

TS inter addition of vectors 20      

   the equation of the line passing through origin and parallel to the vectorTS inter addition of vectors 5isTS inter addition of vectors 21      

  • the  vector equation of the line passing through the points A(TS inter 1A vector a )  and B( TS inter addition of vectors 5 )  is TS inter addition of vectors 23
  • Cartesian equation of the line passing through A ( x1, y1, z1) and  B ( x2, y2, z2) is TS inter addition of vectors 22
  • The vector equation of the plane passing through point A(TS inter 1A vector a ) and parallel to the vectors TS inter addition of vectors 5andTS inter 1A vector c is  TS inter addition of vectors 24
  • The vector equation of the plane passing through the point A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and parallel to the vector TS inter 1A vector c is TS inter addition of vectors 25
  • The vector equation of the plane passing through the points A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and C( TS inter 1A vector c) isTS inter addition of vectors 26

large bar{r}= (1-t)bar{a} + t bar{b}

5.PRODUCT OF VECTORS

TS inter 1A vectors dotproduct title

Dot product (Scalar product): LetTS inter 1A product of vectors 2 are two vectors. The dot product or direct product of TS inter 1A vector a and TS inter 1A vector b  is denoted byTS inter 1A product of vectors 3and is defined as 

  • IfTS inter 1A vector a = 0, TS inter 1A vector b = 0 ⟹ TS inter 1A product of vectors 3  = 0.
  • If TS inter 1A vector a≠0,TS inter 1A vector b ≠ 0 thenTS inter 1A product of vectors 4
  • The dot product of two vectors is a scalar
  • If TS inter 1A product of vectors 2 are two vectors, then

     TS inter 1A product of vectors 1

  • If θ is the angle between the vectorsTS inter 1A product of vectors 2 then. TS inter 1A product of vectors 4

         ⟹    TS inter 1A product of vectors 5

         ⟹ IfTS inter 1A product of vectors 3   > 0, then θ is an acute angle

         ⟹ If  TS inter 1A product of vectors 3  < 0, then θ is obtuse angle 0

          ⟹ If  TS inter 1A product of vectors 3  = 0, thenTS inter 1A vector a  is perpendicular toTS inter 1A vector b

  • IfTS inter 1A vector a is any vector then  TS inter 1A product of vectors 6

Component and Orthogonal Projection:

LetTS inter 1A vector a=TS inter 1A vector OA,TS inter 1A vector b=TS inter 1A vector OB  be two non-zero vectors. Let the plane passing through B (TS inter 1A vector b ) and perpendicular to TS inter 1A vector aintersectsTS inter 1A vector OA

TS inter 1A product of vectors 7

In M, then TS inter 1A vector OM is called the component of TS inter 1A vector b on TS inter 1A vector a

  • The component (projection) vector of TS inter 1A vector b on TS inter 1A vector a is TS inter 1A product of vectors 8
  • Length of the projection (component) =TS inter 1A product of vectors 9
  • Component ofTS inter 1A vector b perpendicular toTS inter 1A vector a = TS inter multiplication of vectors 1

If TS inter 1A vector i,TS inter 1A vector j, TS inter 1A vector k   form a right-handed system of an orthonormal triad, then 

TS inter 1A product of vectors 10

  • If TS inter 1A product of vectors 11 then TS inter 1A product of vectors 3 = a1b1 + a2b2 + a3b3
  • IfTS inter 1A product of vectors 11  then TS inter 1A product of vectors 12

Parallelogram law:TS inter multiplication of vectors 3

In a parallelogram, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of its sides.

TS inter multiplication of vectors 2

In ∆ABC, the length of the median through vertex A is TS inter multiplication of vectors 4

Vector equation of a plane:TS inter multiplication of vectors 10

The vector equation of the plane whose perpendicular distance from the origin is p and unit normal drawn from the origin towards the plane is,TS inter multiplication of vectors 5

•The vector equation of the plane passing through point A (TS inter 1A vector a ) and perpendicular to theTS inter multiplication of vectors 6 isTS inter multiplication of vectors 7

•If θ is the angle between the planes TS inter multiplication of vectors 8 then TS inter multiplication of vectors 9


TS inter VECTORS Cross product 1

Cross product (vector product): Let TS inter 1A vector aandTS inter addition of vectors 5 be two non-zero collinear vectors. The cross product of TS inter 1A vector a  and TS inter addition of vectors 5  is denoted by TS inter 1A vector a×TS inter addition of vectors 5  (read as a cross ) and is defined as TS inter VECTORS Cross product 2

TS inter VECTORS Cross product 3

TS inter VECTORS Cross product 4are orthogonal triad then

TS inter VECTORS Cross product 5

• The vectorTS inter 1A vector a ×TS inter addition of vectors 5 is perpendicular to both TS inter 1A vector a and TS inter addition of vectors 5 and also perpendicular to the plane containing themTS inter VECTORS Cross product 6

• The unit vector perpendicular to bothTS inter 1A vector a and TS inter addition of vectors 5  isTS inter VECTORS Cross product 7

• LetTS inter VECTORS Cross product 8 then TS inter VECTORS Cross product 9

• If TS inter 1A vector aand TS inter addition of vectors 5 are two sides of a triangle then the area of the triangle =TS inter VECTORS Cross product 10

• If A (TS inter 1A vector a ), B ()and C (TS inter 1A vector c )are the vertices of a ∆ABC, then its areaTS inter VECTORS Cross product 12

TS inter VECTORS Cross product 11

• The area of the parallelogram whose adjacent sidesTS inter 1A vector a and TS inter addition of vectors 5   is TS inter VECTORS Cross product 13

• The area of the parallelogram whose diagonals TS inter 1A vector a and TS inter addition of vectors 5   is   TS inter VECTORS Cross product 10

• If A (TS inter 1A vector a ), B (TS inter addition of vectors 5 )and C (TS inter 1A vector c )are three points then the perpendicular distance from A to the line passing through B, C is

TS inter VECTORS Cross product 14


TS inter scallar tripple product 1

LetTS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector c be three vectors, then (TS inter scallar tripple product 3) . TS inter 1A vector c is called the scalar triple product ofTS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector cand it is denoted byTS inter scallar tripple product 2
TS inter scallar tripple product 4
IfTS inter VECTORS Cross product 8TS inter scallar tripple product 21then
TS inter scallar tripple product 5
•In determinant rows(columns) are equal then the det. Value is zero.
•In a determinant, if we interchange any two rows or columns, then the sign of det. Is change.
•Four distinct points A, B, C, and D are said to be coplanar iff TS inter scallar tripple product 6
The volume of parallelepiped:
If TS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector care edges of a parallelepiped then its volume is TS inter scallar tripple product 7
The volume of parallelepiped:
The volume of Tetrahedron withTS inter 1A vector a,TS inter addition of vectors 5 andTS inter 1A vector c are coterminous edges isTS inter scallar tripple product 8
The volume of Tetrahedron whose vertices are A, B, C and D is  TS inter scallar tripple product 9
Vector equation of a plane:
The vector equation of the plane passing through point A (TS inter 1A vector a) and parallel to the vectorsTS inter addition of vectors 5 and TS inter 1A vector cis TS inter scallar tripple product 10
The vector equation of the plane passing through the points A ( TS inter 1A vector a) and B( TS inter addition of vectors 5) and parallel to the vectorTS inter 1A vector c isTS inter scallar tripple product 11
The vector equation of the plane passing through the points A (TS inter 1A vector a), B( TS inter addition of vectors 5) and C(TS inter 1A vector c ) is TS inter scallar tripple product 12
Skew lines:TS inter scallar tripple product 13
The lines which are neither intersecting nor parallel are called Skew lines

The shortest distance between the Skew lines:
If TS inter scallar tripple product 16 are two skew lines, then the shortest distance between them is TS inter scallar tripple product 14

 
If A, B, C and D are four points, then the shortest distance between the line joining the points AB and CD is TS inter scallar tripple product 15

•The plane passing through the intersection of the planesTS inter scallar tripple product 17 is TS inter scallar tripple product 18
the perpendicular distance from point A (a ̅) to the plane TS inter scallar tripple product 19 is TS inter scallar tripple product 20

TS inter 1A vector tripple product 1

Let TS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector c be three vectors, thenTS inter 1A vector tripple product 2 is called the vector triple product ofTS inter 1A vector a,TS inter addition of vectors 5 andTS inter 1A vector c.

TS inter 1A product of four vectors 1

Scalar product of four vectors:

TS inter 1A scalar product of four vectors 1

Vector product of four vectors:

TS inter 1A vector product of four vectors 1


6. TRIGONOMETRY UPTO TRANSFORMATIONS

The word ’trigonometry’ derived from the Greek words ‘trigonon’ and ‘metron’. The word ‘trigonon’ means a triangle and the word ‘metron’ means a measure.

Angle: An angle is a union of two rays having a common endpoint in a plane.

There are three systems of measurement of the angles.

  • Sexagesimal system (British system)
  • Centesimal system (French system)
  • Circular measure (Radian system)

Sexagesimal system: – In this system, a circle can be divided into 360 equal parts. Each part is called one degree (0). One circle = 3600

Further, each degree can be divided into 60 equal parts. Each part is called one minute (‘).

and each minute can be divided into 60 equal parts. Each part is called one second (“)

Sexagesimal system: – In this system, a circle can be divided into 400 equal parts. Each part is called one grade (g). One circle = 400g

Further, each grade can be divided into 100 equal parts. Each part is called one minute (‘).

and each minute can be divided into 100 equal parts. Each part is called one second (“)

Circular measure: Radian is defined as the amount of the angle subtended by an arc of length ’r’ of a circle of radius ‘r’.

One radian is denoted by 1c. One circle = 2πc

 Relation between the three measures:

3600 = 400g = 2 πc

1800 = 200g = πc

TS inrer relation betwee the measurements

Trigonometric Ratios:TS inter trigonometric ratios1

TS inter trigonometric ratios

 Trigonometric identities: –

∗ sin2θ + cos2θ = 1

        1 – cos2θ = sin2θ

        1 – sin2θ = cos2θ

∗ sec2θ − tan2θ = 1

 sec2θ = 1 + tan2θ

tan2θ = sec2θ – 1

(secθ − tanθ) (secθ + tanθ) = 1

TS inter trigonometric identities 1

 ∗  cosec2θ − cot2θ = 1

         co sec2θ = 1 + cot2θ

cot2θ = cosec2θ – 1

(cosec θ – cot θ) (cosec θ + cot θ) = 1

TS inter trigonometric identities 2

• sin θ. cosec θ = 1

sec θ. cos θ = 1

tan θ. cot θ = 1

All Silver Tea Cups Rule:

TS inrer trigonometry all silver tea cups

Note: If 900 ±θ or 2700 ±θ then

‘sin’ changes to ‘cos’; ‘tan’ changes to ‘cot’; ‘sec’ changes to ‘cosec’

‘cos’ changes to ‘sin’; ‘cot’ changes to ‘tan’; ‘cosec’ changes to ‘sec’.

If 1800 ±θ or 3600 ±θ then, no change in ratios.

Values of Trigonometric Ratios:

TS inrer trigonometry ratios

TS inter trigonometric ratios values

Complementary angles: Two angles A and B are said to be complementary angles, if A + B = 900.

supplementary angles: Two angles A and B are said to be supplementary angles, if A + B = 1800.

TS inrer trigonometry periodic fu

Let E ⊆ R and f: E → R be a function, then f is called periodic function if there exists a positive real number ‘p’ such that

  • (x + p) ∈ E ∀ x∈ E
  • F (x+ p) = f(x) ∀ x∈ E

If such a positive real number ‘p’ exists, then it is called a period of f.

TS inrer trigonometry periodic functions1

TS inrer trigonometry COMPOUND AANGLES 1

The algebraic sum of two or more angles is called a ‘compound angle’.

 For any two real numbers A and B

sin (A + B) = sin A cos B + cos A Cos B

sin (A − B) = sin A cos B − cos A Cos B

cos (A + B) = cos A cos B − sin A sin B

cos (A − B) = cos A cos B + sin A sin B

tan (A + B) =

tan (A − B) =ts inter ttrriggonomertty compound angles 2

cot (A + B) =ts inter ttrriggonomertty compound angles 3

⋇ cot (A − B) = ts inter ttrriggonomertty compound angles 4

sin (A + B + C) = ∑sin A cos B cos C − sin A sin B sin C 

cos (A + B + C) = cos A cos B cos C− ∑cos A sin B sin C 

tan (A + B + C) =ts inter ttrriggonomertty compound angles 5

⋇ cot (A + B + C) =ts inter ttrriggonomertty compound angles 6

⋇ sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A

⋇ cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A

ts inter ttrriggonomertty compound angles 11

Extreme values of trigonometric functions:

If a, b, c ∈ R such that a2 + b2 ≠ 0, then

Maximum value = ts inter ttrriggonomertty compound angles 12

Minimum value =ts inter ttrriggonomertty compound angles 13

ts inter trigonometry Multiple and submultiple angles 1

If A is an angle, then its integral multiples 2A, 3A, 4A, … are called ‘multiple angles ‘of A and the multiple of A by fraction likets inter trigonometry Multiple and submultiple angles 2are called ‘submultiple angles.

⋇ sin 2A = 2 sin A cos A =ts inter trigonometry Multiple and submultiple angles 5

⋇ cos 2A = cos2 A – sin2 A

                 = 2 cos2 A – 1

                 = 1 – 2sin2 A

                =ts inter trigonometry Multiple and submultiple angles 6

⋇ tan 2A =ts inter trigonometry Multiple and submultiple angles 3

⋇ cot 2A =ts inter trigonometry Multiple and submultiple angles 4

∎ If ts inter trigonometry Multiple and submultiple angles 7  is not an add multiple of ts inter trigonometry Multiple and submultiple angles 8

⋇ sin A = 2 sints inter trigonometry Multiple and submultiple angles 7  costs inter trigonometry Multiple and submultiple angles 7  =ts inter trigonometry Multiple and submultiple angles 10

⋇ cos A = cos2 ts inter trigonometry Multiple and submultiple angles 7  – sin2 ts inter trigonometry Multiple and submultiple angles 7

                 = 2 cos2 ts inter trigonometry Multiple and submultiple angles 7   – 1

                 = 1 – 2sin2 ts inter trigonometry Multiple and submultiple angles 7

                  =ts inter trigonometry Multiple and submultiple angles 9

⋇ tan A =ts inter trigonometry Multiple and submultiple angles 11

⋇ cot A =ts inter trigonometry Multiple and submultiple angles 12

ts inter trigonometry Multiple and submultiple angles 13

⋇ sin3A = 3 sin A −4 sin3 A

⋇ cos 3A = 4 cos3 A – 3 cos A

⋇ tan 3A =ts inter trigonometry Multiple and submultiple angles 14

⋇ cot 3A =ts inter trigonometry Multiple and submultiple angles 15

⋇ tan A + cot A = 2 cosec 2A

⋇ cot A – tan A = 2 cot 2A

ts inter trigonometry Multiple and submultiple angles 16

TS inter tranformations10

For A, B∈ R

⋇ sin (A + B) + sin (A – B) = 2sin A cos B

⋇ sin (A + B) −sin (A – B) = 2cos A sin B

⋇ cos (A + B) + cos (A – B) = 2 cos A cos B

⋇ cos (A + B) − cos (A – B) = − 2sin A sin B

For any two real numbers C and D

⋇ sin C + sin D = 2sinTS inter tranformations1 cosTS inter tranformations2

⋇ sin C −sin D= 2cosTS inter tranformations1  sinTS inter tranformations2

⋇ cos C + cos D = 2 cosTS inter tranformations1cos TS inter tranformations2

⋇ cos C − cos D = − 2sinTS inter tranformations1   sinTS inter tranformations2   

If A + B + C = π or 1800, then

⋇ sin (A + B) = sin C; sin (B + C) = sin A; sin (A + C) = sin B

⋇ cos (A + B) = − cos C; cos (B + C) = −cos A; cos (A + C) = − cos B

If A + B + C = 900 orts inter trigonometry Multiple and submultiple angles 8  then

⋇ sin TS inter tranformations4  = cosTS inter tranformations5  ; sinTS inter tranformations6    = cosTS inter tranformations7  ; sinTS inter tranformations8    = cosTS inter tranformations9

 

⋇ cos TS inter tranformations4   = sinTS inter tranformations5 ; cosTS inter tranformations6    = sinTS inter tranformations7 ; cos TS inter tranformations8   = sinTS inter tranformations9

If TS inter tranformations3 then

⋇ sin (A + B) = cos C; sin (B + C) = cos A; sin (A + C) = cos B

⋇ cos (A + B) = sin C; cos (B + C) = sin A; cos (A + C) = sin B


 7. TRIGONOMETRIC EQUATIONS

Trigonometric equation: An equation consisting of the trigonometric functions of a variable angle θ ∈ R is called a ‘trigonometric equation’.

The solution of the equation: The values of the variable angle θ, satisfying the given trigonometric equation is called a ‘solution’ of the equation.

The set of all solutions of the trigonometric equation is called the solution set’ of the equation. A ‘general solution’ is an expression of the form θ0 + f(n) where θ0 is a particular solution and f(n) is a function of n ∈ Z involving π.

If k ∈ [− 1, 1] then the principle solution of θ of sin x = k lies in TS inter trigonometric equations1  

General solution of sin x = sin θ is x = nπ + (−1) n θ, n ∈ Z

If k ∈ [− 1, 1] then the principle solution of θ of cos x = k lies in   TS inter trigonometric equations2

General solution of cos x = cos θ is x = 2nπ ± θ, n ∈ Z

If k ∈R then the principle solution of θ of tan x = k lies in TS inter trigonometric equations3  

General solution of tan x = tan θ is x = nπ + θ n ∈ Z

If sin θ = 0, then the general solution is θ = nπ, n ∈ Z

If tan θ = 0, then the general solution is θ = nπ, n ∈ Z

If cos θ = 0, then the general solution is θ = (2n + 1)ts inter trigonometry Multiple and submultiple angles 8 , n ∈ Z

If sin2 θ = sin2 𝛂, cos2 θ = cos2 𝛂 or tan2 θ = tann2 𝛂 then the general solution is 𝛉 = nπ ± θ, n ∈ Z


8.INVERSE TRIGONOMETRIC FUNCTIONS

If A, B are two sets and f: A→ B is a bijection, then f-1 is existing and f-1: B → A is an inverse function.

The function Sin-1: [−1, 1] →TS inter inverse trigonometric functions1 is defined by Sin-1 x = θ ⇔ θ∈ TS inter inverse trigonometric functions1 and sin θ = x

The function Cos-1: [−1, 1] → [0, π] is defined by Sin-1 x = θ ⇔ θ∈ [0, π] and cos θ = x

The function Tan-1: R →TS inter inverse trigonometric functions2  is defined by Tan-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions2  and tan θ = x

The function Sec-1: [−∞, −1] ∪ [1, ∞] →TS inter inverse trigonometric functions5 is defined by Sin-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions5 and sec θ= x

The function Cosec-1: [−∞, −1] ∪ [1, ∞] →TS inter inverse trigonometric functions6   is defined by cosec-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions6 and Cosec θ= x

The function Cot-1: R → (0, π) is defined by Cot-1 x = θ ⇔ θ ∈ (0, π) and cot θ = x

TS inter domain and range of inverse trigonometric functions

Properties of Inverse Trigonometric functions:

Sin-1 x = Cosec-1(1/x) ∀ x ∈ [−1, 1] – {0}

Cos-1x = Sec-1(1/x) ∀ x ∈ [−1, 1] – {0}

Tan-1 x = Cot-1(1/x), if x > 0

Tan-1 x = Cot-1(1/x) −π, if x < 0

Sin-1 (−x) = − Sin-1(x) ∀ x ∈ [−1, 1]

Cos-1 (−x) = π − Cos-1(x) ∀ x ∈ [−1, 1]

Tan-1 (−x) = − Tan-1(x) ∀ x ∈ R

Cosec-1 (−x) = − Cose-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)

Sec-1 (−x) = π − Sec-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)

Cot-1 (−x) =π − Cot-1(x) ∀ x ∈ R 

 (i) If θ∈TS inter inverse trigonometric functions1, then Sin−1(sin θ) = θ and if x ∈ [−1, 1], then sin (Sin−1x) = x

 (ii) If θ∈ [0, π], then Cos−1(cos θ) = θ and if x ∈ [−1, 1], then cos (Cos−1x) = x

 (iii) If θ∈TS inter inverse trigonometric functions2 , then tan−1(tann θ) = θ and if x ∈ R, then tan (Tan−1x) = x

 (iv) If θ∈ (0, π), then Cot−1(cot θ) = θ and if x ∈ R, then cot (Cot−1x) = x

 (v) If θ∈ [0, TS inter inverse trigonometric functions12) ∪ (TS inter inverse trigonometric functions12 , π], then Sec−1(sec θ) = θ and

 if x ∈ (− ∞, − 1] ∪ [1, ∞), then sec (Sec−1x) = x

 (vi) If θ∈ TS inter inverse trigonometric functions6 , then Cosec−1(cosec θ) = θ and

if x ∈ (− ∞, − 1] ∪ [1, ∞), then cosec (Cosec−1x) = x

(i) If θ∈TS inter inverse trigonometric functions1 , then Cos−1(sin θ) = TS inter inverse trigonometric functions7

 (ii) If θ∈ [0, π], then Sin−1(cos θ) =TS inter inverse trigonometric functions7

 (iii) If θ∈TS inter inverse trigonometric functions2 , then Cot−1(tan θ) =TS inter inverse trigonometric functions7

 (iv) If θ∈ (0, π), then Tan−1(cot θ) =TS inter inverse trigonometric functions7

 (v) If θ∈ TS inter inverse trigonometric functions5, then Cosec−1(sec θ) =TS inter inverse trigonometric functions7

 (vi) If θ∈TS inter inverse trigonometric functions6 , then Sec−1(cosec θ) =TS inter inverse trigonometric functions7

  1. Sin1x = Cos( TS inter inverse trigonometric functions8)if 0 ≤ x ≤ 1 and Sin1x =− Cos1 ( TS inter inverse trigonometric functions8) if −1 ≤ x ≤ 0
  2. Sin1x = Tan1TS inter inverse trigonometric functions9 if x ∈ (−1, 1)
  3. Cos1x = Sin1 (TS inter inverse trigonometric functions13) if x ∈ [0, 1] and Cos1x = π − Sin1 (TS inter inverse trigonometric functions13)  if x ∈ [−1, 0]
  1. Tan1x = Sin1TS inter inverse trigonometric functions10 = Cos−1 TS inter inverse trigonometric functions11or x > 0

Cos−1 x + Sin−1x = TS inter inverse trigonometric functions12  ∀ x ∈ [−1, 1]

Tan−1 x + Cot−1x =TS inter inverse trigonometric functions12  ∀ x ∈ R

Sec−1 x + Cosec−1x = TS inter inverse trigonometric functions12 ∀ x ∈ (−∞, −1] ∪ [1, ∞) 

Sin−1 x + Sin−1y = Sin−1(x TS inter inverse trigonometric functions14 + yTS inter inverse trigonometric functions13  ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 ≤ 1

                                    =π− Sin−1(x TS inter inverse trigonometric functions14 + y TS inter inverse trigonometric functions13 ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 > 1

Cos−1 x + Cos−1y = Cos−1(x y −TS inter inverse trigonometric functions13  TS inter inverse trigonometric functions14 ) if 0 ≤x, y ≤ 1and x2 + y2 ≥ 1

                                    =π− Cos−1(x y −TS inter inverse trigonometric functions13 TS inter inverse trigonometric functions14  ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 < 1

Tan−1 x + Tan−1y = Tan−1TS inter inverse trigonometric functions15  if x > 0, y> 0 and xy < 1

                                    =π + Tan−1 TS inter inverse trigonometric functions15 if x > 0, y> 0 and xy > 1

                                    =   Tan−1 TS inter inverse trigonometric functions15if x < 0, y< 0 and xy > 1

                                  = −π + Tan−1TS inter inverse trigonometric functions15  if x < 0, y< 0 and xy < 1

Tan−1 x − Tan−1y = Tan−1 TS inter inverse trigonometric functions16 if x > 0, y> 0 or x < 0, y< 0

2 Sin−1 x = Sin−1 (2x ) if x≤TS inter inverse trigonometric functions17

                       = π− Sin−1 (2x ) if x >TS inter inverse trigonometric functions17

2 Cos−1 x = Cos−1(2x2 – 1) if x ≥TS inter inverse trigonometric functions17

                        =Cos−1(1–2x2) if x <TS inter inverse trigonometric functions17

2 Tan−1 x = Tan−1 TS inter inverse trigonometric functions23 ifTS inter inverse trigonometric functions18 < 1

                         = π + Tan−1 TS inter inverse trigonometric functions23 ifTS inter inverse trigonometric functions18 ≥ 1

                         = Sin−1 TS inter inverse trigonometric functions19 if x ≥ 0

                         = Cos−1 TS inter inverse trigonometric functions20 if x ≥ 0

3Sin−1x = Sin−1(3x – 4x3)

3Cos−1x = Cos−1(4x3 – 3x)

3Tan−1x = tan−1TS inter inverse trigonometric functions21


9.HYPERBOLIC FUNCTIONS

TS inter Hyperbolic functions 1

The function f: R→R defined by f(x) =  ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by sinh x.

∴ sinh x =TS inter Hyperbolic functions 2

Similarly,

cosh x = TS inter Hyperbolic functions 3 ∀ x ∈ R 

tanh x = TS inter Hyperbolic functions 4 ∀ x ∈ R 

coth x =TS inter Hyperbolic functions 5  ∀ x ∈ R

sech x = TS inter Hyperbolic functions 6  ∀ x ∈ R

cosech x = TS inter Hyperbolic functions 7  ∀ x ∈ R

Identities:

cosh2x – sinh2 x = 1

    cosh2x = 1 + sinh2 x

    sinh2 x = cosh2 x – 1

sech2 x = 1 – tanh2 x

    tanh2 x = 1 – sesh2 x

cosech2 x = coth2 x – 1

     coth2 x = 1 + coth2 x

Addition formulas of hyperbolic functions:

sinh (x + y) = sinh x cosh y + cosh x sinh y

sinh (x − y) = sinh x cosh y − cosh x sinh y

cosh (x + y) = cosh x cosh y + sinh x sinh y  

cosh (x − y) = cosh x cosh y − sinh x sinh y  

tanh (x + y) = TS inter Hyperbolic functions 8

tanh (x − y) = TS inter Hyperbolic functions 9

coth (x + y) =TS inter Hyperbolic functions 10  

sinh 2x = 2 sinh x cosh 2x = TS inter Hyperbolic functions 11

cosh 2x = cosh2x + sinh2 x = 2 cosh2x – 1 = 1 + 2 sinh2x =TS inter Hyperbolic functions 12

tanh 2x =TS inter Hyperbolic functions 13

sinh 3x = 3 sinh x + 4 sinh3x

cosh 3x = 4 cosh3 x – 3 cosh x

tanh 3x = TS inter Hyperbolic functions 14

Inverse hyperbolic functions:

Sinh−1x =TS inter Hyperbolic functions 15  ∀ x ∈ R

Cosh−1x = TS inter Hyperbolic functions 21  ∀ x ∈ (1, ∞)

Tanh−1x = TS inter Hyperbolic functions 16   ∀ TS inter inverse trigonometric functions18< 1

Coth−1x = TS inter Hyperbolic functions 17   ∀ TS inter inverse trigonometric functions18> 1

Sech−1x = TS inter Hyperbolic functions 18   ∀ x ∈ (0, 1]

Cosech−1x = TS inter Hyperbolic functions 19   if x < 0 and x ∈ (−∞, 0)

                         = TS inter Hyperbolic functions 18  if x > 0

TS inter Hyperbolic functions 20


10. PROPERTIES OF TRIANGLES

 In ∆ABC,TS inter Properties of triangles 1

Lengths AB = c; BC = a; AC =b

Area of the tringle is denoted by ∆.

Perimeter of the triangle = 2s = a + b + c

A = ∠CAB; B = ∠ABC; C = ∠BCA.

R is circumradius.

Sine rule:

In ∆ABC,

TS inter Properties of triangles 2

 ⟹ a = 2R sin A; b = 2R sinB; c = 2R sin C

Where R is the circumradius and a, b, c, are lengths of the sides of ∆ABC.

Cosine rule:

In ∆ABC,

a2 = b2 + c2 – 2bc cos A    ⟹cos A = TS inter Properties of triangles 3

b2 = a2 + c2 – 2ac cos B    ⟹ cos B = TS inter Properties of triangles 4

c2 = a2 + b2 – 2ab cos C    ⟹ cos A = TS inter Properties of triangles 5

projection rule:

In ∆ABC,

a = b cos C + c cos B

b = a cos C + c cos A

c = a cos B + b cos A

Tangent rule (Napier’s analogy):

In ∆ABC,

TS inter Properties of triangles 6

Half angle formulae and Area of the triangle:TS inter Properties of triangles 8

In ∆ABC, a, b, and c are sides

TS inter Properties of triangles 7   and area of the triangle TS inter Properties of triangles 13

1.Half angle formulae: –

TS inter Properties of triangles 9

TS inter Properties of triangles 10

TS inter Properties of triangles 11

TS inter Properties of triangles 12

2.Formulae for ∆: – 

∆ = ½ ab sinC= ½ bc sin A=½ ac sin B

  TS inter Properties of triangles 13  where TS inter Properties of triangles 7

   = 2R2sin A sin B sinC

   = r.s

   =TS inter Properties of triangles 14

  =TS inter Properties of triangles 15

In circle and Excircles of a triangle:TS inter Properties of triangles 16

⋇The circle that touches the three sides of an ∆ABC internally is called ‘incircle’. The centre of the incircle is ‘I’ and the radius is ‘r’.

Formulae for ‘r’: –

r = TS inter Properties of triangles 27

 = (s – a) tanTS inter Properties of triangles 17  = (s – b) tanTS inter Properties of triangles 18  = (s – c) tanTS inter Properties of triangles 19

 = 4R sinTS inter Properties of triangles 17sinTS inter Properties of triangles 18 sinTS inter Properties of triangles 19

 =TS inter Properties of triangles 20

The circle that touches the side BC internally and the other two sides AB and AC externally is called the ‘Excircle’ opposite to the angle A. Its centre is I1 and the radius is r1. A TS inter Properties of triangles 21triangle has three ex circles. The remaining circles centre and radius are respectively I2, r2 and I3, r3.

  Formulae for ‘r1’: –

r1 = TS inter Properties of triangles 22

 = s tan TS inter Properties of triangles 17 

= (s – b) cotTS inter Properties of triangles 19  = (s – c) cotTS inter Properties of triangles 18

 = 4R sinTS inter Properties of triangles 17  cosTS inter Properties of triangles 18  cosTS inter Properties of triangles 19

 =TS inter Properties of triangles 23

Formulae for ‘r2’: –

r2 = TS inter Properties of triangles 24

= s tanTS inter Properties of triangles 18   

 = (s – c) cotTS inter Properties of triangles 17  = (s – a) cotTS inter Properties of triangles 19

= 4R cosTS inter Properties of triangles 17  sinTS inter Properties of triangles 18 cosTS inter Properties of triangles 19

 =TS inter Properties of triangles 25 

Formulae for ‘r3’: –

r3 = TS inter Properties of triangles 28

 = s tanTS inter Properties of triangles 19

= (s – a) cot TS inter Properties of triangles 18 = (s – b) cotTS inter Properties of triangles 17

 = 4R cosTS inter Properties of triangles 17  cosTS inter Properties of triangles 18  sinTS inter Properties of triangles 19

 =TS inter Properties of triangles 26


Visit my Youtube Channel: Click on Below Logo

My Youtube channel Logo

 

error: Content is protected !!