MEMORISE A B AND C D FORMULAS IN TRIGONOMETRY IN AN EASY MANNER

Hello Friends 

welcome to this post of learning trigonometric formulas with me .Most of the Students or Mathematics Learner ,most of the time confuse to remember or memorise A  B and C  D formulas,They  mixed A B and C D formulas with each other and could not reproduce what they have learnt . So today we going to learn new techniques to learn "How to memorise AB and CD formulas" forever. Before this we must have knowledge of different trigonometric values of different angles in different quadrants.

First of all have a quick look at some of  these formulas .

To clear your  all doubts on   " How to Calculate Different Trigonometric values in different quadrants "  in an easy Method. click on the  above  links  .

 

Tricks to Learn    A  B   Formulae  For  sine  angles

When angles are added   i. e  Sin  ( A+B )  

When Angles are added and then their Trigonometric Ratios is taken , and if we have to take the  Sine of  added angles, then it can be done like this.

Start with  sine of angle A  and multiply it with cosine of  angle B and in other part i. e.  After  +ve sign      Start with Cosine of angle A and multiply with Sine of angle B. i.e. start with sine and ends with sine and in middle both the terms are cosine ,and angles start  A then B again A then again B.

Sin (A+B) = Sin A Cos B + Cos A Sin B

When angles are subtracted    i. e  Sin  ( A-B )  

When Angles are subtracted and  their Trigonometric Ratios is taken , and if we have to take the  Sine of  subtracted  angles, then it can be done like this

    
Start with  sine of angle A  and multiply it with cosine of  angle B and in other part i. e.  After  +ve sign    Start with Cosine of angle A and multiply with Sine of angle B. i.e  start with sine and ends with sine and in middle both the terms are cosine ,and angles start  with A then B again A then again B.

Sin (A - B) = Sin A Cos B - Cos A Sin B

Tricks to Learn    A  B   Formulae For  Cosine  angles

When angles are added   i. e  Cos  ( A+B )  

When Angles are added and then their Trigonometric Ratios is taken , and if we have to take the  Cosine  of  added angles, then it can be done like this.

Start with  cosine of angle A  and multiply it with cosine of  angle B and in other part i. e.  After  -ve sign      Start with Sine of angle A and multiply with Sine of angle B. i.e. 1^st   and 2^nd terms are   cosine and  3^rd and 4^th terms    are sine , Angles start with   A then B again A then again B.

"Here  Sum of cosine of  Two angles  is equal to difference of  product of  cosines of both the angles    and product of sine of both the angles ".

Cos (A+B) = Cos A Cos B - Sin A Sin B

When angles are subtracted    i. e  Cos  ( A-B )  

When Angles are subtracted and  then their Trigonometric Ratios is taken , and if we have to take the  cosine of  subtracted  angles, then it can be done like this.

Cos (A - B) = Cos A Cos B + Sin A Sin B

Start with  cosine of  angle A  and multiply it with cosine of  angle B and in other part i. e.  After  +ve sign      Start with Sine of angle A and multiply with Sine of angle B. i. 1^st   and 2^nd terms are   cosine and  3^rd and 4^th  terms    are sine , and angles start with   A then B again A then again B.

"Here  Difference  of cosine of  Two angles  is equal to the  Sum  of  product of  cosines of both the angles    and product of sine of both the angles" .

Want to Learn WHAT IS SET, TYPES OF SETS ,UNION ,INTERSECTION AND VENN DIAGRAMS .

How  to  Memorise     C D   Formulae

To learn C D formulae 

Step 1 

Place 2 for all four formulae and  take Trigonometric Ratio of 1st angle for all four formulae which  is (C+D)/2 and again  trigonometric Ratio of  2nd angle which  is (C-D)/2.

Step 2.1 

For addition of Sine Formula start with sine of 1st angle as mentioned in step 1 and multiply it with cos of  2nd angle as mentioned in step 1.

Step 2.2 

For subtraction of Sine Formula start with cosine of 1st angle as mentioned in step 1 and multiply it with sine of  2nd angle as mentioned in step 1.

Step 3.1

For addition of cosine Formula start with cosine of 1st angle as mentioned in step 1 and multiply it with cosine  of  2nd angle as mentioned in step 1.

Step 3.2

For subtraction of cosine Formula start with sine of 1st angle as mentioned in step 1 and multiply it with sine   of  2nd angle as mentioned in step 1,and do not forget to multiply it with -ve sign.

or  

If you do not want to multiply it with -ve sign  ,then you can change 2nd angle (D-C)/2 instead of (C-D)/2

How to Memorise A   B and C   D  formulas  easily ,watch this video 

Want to learn MATRIX , DIFFERENT TYPES OF MATRICES AND DETERMINANTS .


Want to learn HOW TO SOLVE LINEAR EQUATIONS OF TWO AND THREE VARIABLES || MATRIX METHOD TO SOLVE SYSTEM LINEAR EQUATIONS .

Conclusion

Thanks for devoting your valuable time for the post Easy Tricks to Memorise A B and C D Formulae in Trigonometry and trigonometry's shortcut formulas of this blog. If you found this this blog/post of your concern, Do Follow me on my blog and share this post with your friends . We shall meet again in next post ,till then Good Bye.

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HOW TO SOLVE LINEAR EQUATIONS OF TWO AND THREE VARIABLES BY MATRIX METHOD

Today we are going to learn, how to solve system of linear equations of 2 and 3 variables using matrix method. So using matrix method to solve system of linear equation , we must know some topics such as co-factor of element, Transpose of matrix , Ad joint of a Matrix, Multiplication of two Matrices ,Determinant value of a Matrix ,Inverse of a matrix etc. Let us understand the process of finding the solution of system of linear equations with the help of some examples.

To Solve the system  of Linear Equations using 2×2 Matrix Method

x - 5y = 4

2x + 5y = −2

Writing this system of equation in Matrix form

AX = B 

where X = A-1 B------------------------(1)


And A-1 = (1/Det A ) ( Ad joint A)



Where

We need the inverse of   A , which we write as   A^-1 

To find the inverse 1st find out Co-factor Matrix of A

Ad joint  A  =   (     5    5​ -2   1   ​)

Co factor   A  =   (     5    −2​ 5   1   ​)

As we know the Ad joint Matrix of any matrix can be found by taking the transpose of the Co Factor matrix.

Now let us find the determinant of  A

∣A∣ = 5 − (-10) = 15 , which is non Zero,


Therefore  A^-1  Exists     ,So

Now putting the value of inverse of Matrix A in equation (1)

Now putting the elements of Matrix X , and 

By the equality of two Matrices ,their elements in respective positions are equal to each others,

Hence x= 2/3  and y = -2/3

How to check the correctness of a solution

put x=  2/3  and y = -2/3  in the given  equations  

x - 5y = 4 implies 2/3 - 5(-2/3) = 2/3+10/3 = 4

and 2x + 5y  = -2 implies 2*(2/3) + 5(-2/3) = 4/3-10/3 = -6/3= -2

therefore both equations are satisfied by these values of 'x' and 'y'

To Solve the system of Linear Equations using 3×3 Matrix Method

 x  - y + z   = 4
2x + y - 3z  = 0
 x + y + z    = 2

Converting this system of Linear equations into Matrix Form

AX    =  B 

X  =    A^-1  B ----------------------------------(1)
Where

Now we have to find the   A^-1 . , and it will exist if its determinants value |A| must be non Zero. Let us find |A|

    




 |A|= 1{1×1-(-3)×1}-(-1){2×1-(3)×1}+1(2×1-1×1)

|A|= 1 {1+3}+1 {2+3}+1 {2-1}

|A|= 4+5+1 = 10

Since |A| is non Zero , Therefore inverse of A exists .

We have to find the Co factors of all the elements of matrix A . Let us find out the co factors of all the elements of matrix row wise.

Co-Factors of 1st Row

     

            co-factor of A11  =   4
            co-factor of A12  =  -5
            co-factor of A13  =   1

Co-Factors of 2nd Row

            co-factor of A21  =   2 

            co-factor of A22  =   0
            co-factor of A23  =  -2

Co-Factors of 3rd Row

              co-factor of A31  =  2
              co-factor of A32  =  5
              co-factor of A33  =  3

Now co Factor matrix of A can be written as follows




Now to find the Ad joint of this Matrix, Take the transpose of this Matrix, 

As we know the Inverse of a Matrix  is the scalar  multiplication of Ad Joint Matrix of Matrix A and reciprocal of Determinant value of the Matrix A.

Putting the values of Inverse of A and Matrix B in equation (1), So after Multiplication of these two matrices ,we get

$\backslash displaystyle=\{\backslash left(\backslash begin\{matrix\}\{22\}\backslash \backslash -\{16\}\backslash \backslash -\{16\}\backslash end\{matrix\}\backslash right)\}$

Now putting the values of elements  of   Matrix X, and equating the elements in their respective positions .

Now using the property of equality of two Matrices , All the elements in their respective position are equal to each other , we get

 x = 2  , y = -1, z = 1  is the solution of the system of linear Equations.

Conclusion

In this post I  discussed the method of solving the System of linear Equations with the help of Matrix Method , If you liked the post please share it with your friends , And in case of any improvement please make use of Comment Box . To keep on supporting me follow my Blog . Thanks for your valuable time to read this post .We shall meet you in next post ,till then BYE.

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HOW TO MULTIPLY TWO MATRICES || PRODUCT OF TWO MATRICES

In this post we are  going to discuss  multiplication of two matrices , i.e. when two matrices can be multiplied with each other or when two matrices are NOT eligible for multiplication, what are the necessary conditions for the multiplication of two matrices to be multiplied.

Conditions for Matrix Multiplication

Before multiplication of two matrices we have to check whether multiplication is possible or not , If it is possible then matrices will be multiplied to each other. Necessary condition for multiplication of two matrices is if " The number of columns in the first matrix is the same as the number of rows in the second matrix ". We must know  different types of matrices ,Rows and Column.

Note : The commutation may or may not be possible for multiplication of matrices, That is  in some case AB = BA but In general AB is not equal to BA.

Example 1

1. Multiplication of 2 × 3 matrix with 3 × 4 matrix is possible as number of columns in 1st Matrix is equal to numbers of rows in 2nd Matrix and Resultant matrix will be of 2 × 4 order .

2. Multiplication of 5 × 1 matrix with 1 × 2 matrix is also possible as it gives 5× 2 matrix as resultant Matrix.

3. Multiplication of 4 ×3 matrix with  2 × 3 matrix is NOT possible. Because red colour numbers 3 and 2 do not match .

How to Multiply Matrices 

If condition for matrix multiplication is satisfied, then below is the method of multiplication for two matrices .

Let's take  two matrices A and B  of  2 × 3 and 3×2  orders respectively. The resultant matrix will be  2 × 2 matrix. we start across the 1st row of the first matrix multiplying down the 1st column of the second matrix, element by element. Then we add the resulting products. Our product will be written in position a11 (top left) of the answer matrix which will be equal to   "au+bw+cy".

We shall repeat similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result will be written  in position a12 which will comes out "av+bx+cz".

Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result will be placed in position a21  as   "du+ew+fy" .Finally, Multiply the 2nd row of the first matrix and the 2nd column of the second matrix to get the product as "dv+ex+fz" The result is placed in position a22. So the resultant matrix will be written as :--

Let us take one  example to Multiply 2×3 and 3×2 Matrices ,The order of resultant matrix will be 2 × 2.

A11 element will be 4×(-3)+1×5+4×6 =  -12+5+24 =17

A12 element will be   4×1+1×6+4×4  =  4+6+16 = 26

A21 element will be  2×(-3)-5×5+7×6 = -6-25+42 = 11

A22 element will be  2×1+(-5)×6+7×4 = 2-30+28 = 0

Since the resultant Matrix  2× 2   as follows

A11  element will be 8×(-2)+9×4      = -16 +36 =20

A12   element will be 8×3+9×0            = 24+ 0 = 24

A21  element will be 5×(-2)+(-1)×4   = -10-4 = -14

A22  element will be 5×3+(-1)×0         = 15+0 = 15
so the resultant matrix will be written as 

To find AB

Since Numbers of columns (3) in 1st matrix (Here A Matrix ) is not equal to numbers of Rows (1) in 2nd Matrix (Here Matrix B ). Therefore matrix multiplication is NOT possible in this case.

To Find BA

Since Numbers of columns (3) in 1st matrix (Here B Matrix ) is equal to numbers of Rows (3) in 2nd Matrix (Here A Matrix ). And resultant Matrix shall be 1×3 order . Therefore matrix multiplication is possible in this case and can be calculated in the following manner :-

BA = ( 4*2+3*3+8*3      4*1+3*(-5)+8*2        4*(-7)+3*0+8*(-1) )

BA = ( 8+9+24    4-15+16    -28+0-8 )

BA = ( 41    5     -36 )

Therefore AB is not equal to BA
This video can better demonstrate the multiplication of two matrices .

Important Note

And what about the matrix multiplication of AB and BA of the matrices given below whose order are 4×3 and 3×3 respectively . 

Then Matrix Multiplication BA is not possible as the numbers of columns(3) in matrix  B is   not  equal to the numbers of rows (4) in  matrix A .

But If we want to Multiply the matrix A with Matrix B as AB then it is possible as the numbers of columns(3) in A matrix is equal to the numbers of rows(3) in B matrix, so it can be calculated as explained earlier. 

It means the order of the matrices play very important role to decide whether matrix multiplication is possible or not. And if possible it is again the order of both the matrices to decide what will be the order of the resultant matrix.

Therefore in  multiplication of matrices of 4×3 and 3×3 .The order of resulting matrix will be 4×3 ,The outer most numbers ( Marked red ). 

Very Important 

If we have two matrices A and B of order 1×3  and 3×1 respectively ,  and if we have find their product AB  and BA , then  

In  1st case  where we have to find AB , let us check its multiplication would be possible or not , as number of columns (3) in 1st matrix is equal to numbers of rows (3) in 2nd Matrix , so AB would be possible and order of resulting matrix would be 1×1 and it can be found using matrix multiplication .

And if we have find their product BA , then would it be possible or not ? let us check .....

In 2nd case where we have to find BA , let us check its multiplication would be possible or not , as number of columns (1)  in 1st matrix is equal to numbers of rows (1) in 2nd Matrix , so AB would be possible and order of resulting matrix would be 3×3 and it can be found using matrix multiplication .

Read this one also  HOW TO UNDERSTAND  BINARY OPERATIONS  ,RELATIONS AND FUNCTIONS   ||  COMMUTATIVE || ASSOCIATIVE

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Conclusion 

This post was about multiplication of two Matrices, Eligibility conditions for product of matrices, Thanks for giving your valuable time to this post . If you liked this post,Please share your valuable opinions about this post. See you in next post ,till then Bye.

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$$\displaystyle{A}={\left(\begin{matrix}-{2}&{1}&{7}\\{3}&-{1}&{0}\\{0}&{2}&-{1}\end{matrix}\right)}    
$$\displaystyle{A}={\left(\begin{matrix}-{2}&{1}&{7}\\{3}&-{1}&{0}\\{0}&{2}&-{1}\end{matrix}\right)}   

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MATRIX , DIFFERENT TYPES OF MATRICES AND DETERMINANTS

Hello every one ,Welcome once again, Today we are going to discuss What is the  matrix, elements of a matrix, order of a matrix, different types of matrix , transpose of matrix, ad joint of matrix, determinant  and  how to find the determinant of a matrix .

What is  Matrix

Matrix definition

A Matrix is a set of elements ( Numbers ) arranged in a particular numbers of Rows and columns in a rectangular table. Matrices inside parentheses ( ) or brackets [ ] is the matrix notation.

Here we have an examples of Matrices . 

The elements which are written in horizontal lines are called Row and elements which are written in Vertical lines are called Column. In the matrix A above ↑ the elements 5, 3, -2 are written in Row wise whereas the elements 5,4,3 are written columns wise, Similarly the elements 4,-1,7 are written in Row wise whereas the elements 3,-1,4 are written columns wise .

Order Of a Matrix

If any matrix have "m" number of rows and "n" number of columns , then "m×n" will be the order of that matrix. It is also called matrix dimensions . For  matrices  given below ,

[      5      6    -4    2    ]   This  matrix has 1×4 order,

    3

[  8 ]  Matrix has 3×1 order ,

   -2 

  [  -1 ] Matrix has 1×1 order.

And the matrices A,B,C and D  above have 3×3 ,2 ×2 , 3×4 and 3×2  respectively, as Matrix A has 3 rows and 3 columns, matrix B has 2 rows and 2 columns, Matrix C has 3 Rows and 4 Columns Similarly Matrix D has 3 rows and 2 columns.

Elements of a matrix

The Elements are entries or numbers used in the matrix . These are denoted by aij , where i is the row's number and j is the column's number in which the element is lying.

Consider the matrix A above which has elements  5 ,3,-2 in the 1st rows 4, -1,7 in the 2nd row and   3,4, -1 in the 3rd rows.

Types of matrix

Equality of Matrices

Two matrices are said to be equal ,if they have same order and same  elements at corresponding positions.

                                                                                                                                                              

If we compare matrices A,B,C and D given above then the matrices A and B may be equal to each other provide x = 6 and y = 3, Similarly Matrices C and D may be equal to each other provided x = 8 ,y = 7 and z = 3 .

Note that Matrices A and C can not be equal to each other, because A and C have not same order. Similarly B and D can not be equal to each other as B and D have different order.

Square Matrix

Any Matrix which have equal numbers of rows and columns,Then   That Matrix is called Square matrix .  A,B,C,D,E and F Matrices given below and above  are the examples of Square Matrix.

Zero , Scalar And Diagonal Matrix

Zero Matrix

If all the elements of a matrix are equal to zero then the matrix  is called Null Matrix ,It is also called Void Or Null Matrix. Matrices A  ,B and C given below are the example of Zero Matrices, As all these matrices have all the elements  zeros. so these matrices are Zero Matrices.

Diagonal Matrix

A Diagonal matrix is a square matrix in which all the elements are zero except diagonal elements . Matrices D ,E and F given below are examples of   Diagonal matrix of  2 × 2 Diagonal matrix and 3×3 orders . 

Scalar Matrix

A Diagonal matrix  matrix in which all the elements are zero and diagonal elements are equal to each others. Matrices A,B,C,D,E and F are examples of Scalar Matrices .Given Matrices D ,E and F given below are examples of   Diagonal matrix of  2 × 2 Diagonal matrix ,3×3  and 3×3 orders respectively . 

Identity Matrix

The Identity Matrix, symbolised as I, is a square matrix. where all the elements are 0 except the diagonal elements  , and all the diagonal  elements equal to one. Identity matrix is also called Unit Matrix.

Above two  examples are  3 × 3 and 2×2 Identity matrices respectively . 1st matrix is an example of  3 × 3 matrix and 2nd is an example of 2×2 Identity matrix. Because both the matrices have diagonals elements one and remaining elements  zeros. Identity Matrices and Unit Matrices are also the examples of Scalar Matrices

Row matrix

Any matrix which has only one row is called Row Matrix.  The  Row matrix may have any numbers of  columns .

[      5      6    -4    2    ] ,     [  2    5   ] ,    [-4    6    -6    0 ]

,[      7      4     3    -3    ]

Column Matrix

Any matrix which has only one column is called Column Matrix. The  Column matrix may have any numbers of Rows.

  

    3

[  8 ] ,      [  -1 ] 

    -2

Transpose of matrix

The transpose of any matrix is obtained by transfer of Rows into Columns and vice versa. i.e transforming 1st row to 1st column and transforming 2nd row to 2nd column and so on for 3rd ,4th and 5th rows and columns.

To find the transpose of any matrix , Shift all the elements of all the Rows into respective Columns. Given below matrices have their transpose written on the right side of them.

It  can be seen that in all the examples given above all the elements of particular row have been changed to corresponding column. So A' is the transpose of  Matrix A ,similarly B' is the transpose of  Matrix B .

Determinant of a matrix

A determinant is a square array of numbers which represents a certain sum of products. we can find out a fixed value of determinant, consider an example of a 3 × 3 determinant , it has 3 rows and 3 columns).

Minor of an Element

The minor of an element aij is the determinant obtained by deleting the ith row and the jth column and is denoted by Mij .

Co factor of an Element

The co-factor of an element aij is the determinant obtained by deleting the ith row and the jth column and is multiplied by(-1)^(i+j)and is denoted by Cij. and Cij= (-1)^(i+j)×Mij

We can find the value of a 2 × 2 determinant   as follows  ,1st we multiply the  top left × bottom right first then subtract from it the product of top right element and left bottom. or 

(1st element in 1st row) × [Its Co factor] - (2nd element in 2nd row)× [Its Co factor].

 

Determinant value of above matrix is 4×5-(-3)×5= 20+15=35

How to find the determinant of a 3x3 matrix  

To find the determinant value 3×3 determinant .

1st element in 1st row ×[Its Co factor]-2nd element in 2nd row[Its Co factor]+3rd element in 3rd row [Its Co factor ].

Let us calculate the determinant value of 3×3 matrix  given above 

=2[(-4×-7) - (2×5)] - (-1) [3×(-7) - (2×5)]+ 5[(3×5) - (-4)×5]

=2[28-10]+1[-21-10]+5[15+20]

=36 - 31+75

=80

Ad joint of a Matrix

The transpose of a co factor matrix of any matrix is called ad joint of the Matrix.To find the ad joint Matrix ,1st find the co factors of  all the elements of given Matrix A.
Co factors of 1st row of  Matrix A are 7 and 5
Co factors of 2nd row of Matrix  A are -2 and -3

,then form the Matrix of these  co factors and name it Co factor  matrix, and after that take the transpose of the co factor matrix so formed.Then we have  transpose of the Ad joint of Matrix.

Conclusion

Thanks for giving your valuable time to the post "What is matrix, element of matrix, dimension of matrix, different types of matrix, transpose of matrix, ad joint of matrix, what is a determinant ,  determinant of 3x3 matrix ,  determinant of a 2x2 matrix " of this blog .If you found this post helpful to you , then share it with yours friends and family members . Also follow me on my blog for notifications of next posts.We shall meet again in next interesting and educating post , till then Good Bye. Take care ....

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