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HOW TO PROVE TRIGONOMETRIC IDENTITIES || TRIGONOMETRY

How to prove  Identity cos 6x = 32cos6 x - 48cos4 x   + 18cos2 x  - 6cos2 x  - 1

Proof


L.H.S. = cos 6x =  cos (3.2x) 

Now using the result cos 3θ = 4cos3 θ - 3 cos θ
     
           = 4cos3 2x - 3 cos 2x 
Now using the result  1 + cos 2θ = 2 cos2  θ
  
                                   ⇒       cos 2θ = 2 cos2 θ -1

= 4 {2 cos2 x -1 }3 - 3 {cos2 x  -1}

Now using the result {a - b }3 = {a}3 - b }3  -  3{a }2 b   +3(a) b2

  cos 6x    =4[  {2cos2 x }3 -{ 1 }3  -  3{2cos2 x  }2 1   +3(2cos2 x) 12 ] - 3 × {cos2 x  -1}


cos 6x  =   4[ 8cos6 x  - 1 - 12cos4 x  + 6cos2 x]  - 3{2cos2 x-1}

 cos 6x  = 32cos6 x  - 4 - 48cos4 x  + 24cos2 x  - 6cos2 x + 3

cos 6x   = 32cos6 x - 48cos4 x  + 18cos2 x  - 6cos2 x  - 1






Prove the Identity 

tan (2x) =  2tan x  1 - tan2 x 

Proof

We know that 


tan (A+B) =  tan A +  tan B1 - tan A tan B 

Put A = B  = x in above formula . then it becomes

tan (x+x) =  tan x +  tan x1 - tan x tan x 



tan (2x) =  2tan x  1 - tan2 x 




Prove that sin 2x = 2sin x cos x

Proof
As we know that sin (A + B) = sin A cos B + cos A sin B..  ...(1)

Put A = B  = x in ...   (1)

sin (x + x) = sin x cos x + cos x sin x

sin (2x) = sin x cos x +  sin x cos x

sin (2x) = 2 sin x cos x


Prove that cos 2x = cos2 x - cos2 x

Proof

As we know that cos (A + B) = cos A cos B - sin A sin B..  ...(1)
Put X = A = B in (1) , we get

cos (x + x) = cos x cos x - sin x sin x
cos 2x = cos2 x - sin2 x   

 
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SOME OF MY PUBLISHED POSTS ARE APPEARING HERE IN CHRONOLOGICAL ORDER

In this post I have shown here my some of previous published posts in chronological order.Let us discuss them one by one.

Example :   To Find the Square of Two Digits Numbers Ending with 5



Here I shall show  how square of 35 can be  found quickly.

Step 1 


when unit/(Right most)  place of a number ends with 5 
then place 25 as last two digits (i. e Right most ) of the result.

Step 2 



Take the digit which is on the unit's place and multiply it with its successor (next digit from it), Here in this case the 1st two digits are 12, the product of 3 and its  successor 4. 

Step 3

The result will be  four digit numbers whose first two digits are answer  of step 2 and next two digits are answer of step 1

The square of  35 = (3×4)(25)
                                        = 1225


Example


To find Square of   65   and many more Examples


To find the transpose of matrix 


HOW TO FIND THE TRANSPOSE OF  MATRIX
1st of all shift all the elements which are in 1st row to 1st column as
5
5
2
,then shift the elements which are in      2nd row to 2nd column as  

-1
-3
 7
similarly  shift all the elements which are in 3rd row to 3rd column as
4
2
8

And the matrix so obtained is the transpose matrix. We can  check that same colour row have been  transformed to same colour column
        
HOW TO FIND THE TRANSPOSE OF  MATRIX  WITH AN EASY   METHOD


Now we shall take this  example to find the transpose Matrix 
HOW TO FIND THE TRANSPOSE OF  MATRIX


How to find the multiplication using short cut Method   52324 × 11= ?


1 Place zero at right end of the multiplicand like this 523240

2 Now add 0 to its neighbour 4 as 0 + 4 = 4
3 Now add 4 to its neighbour 2 as 4 + 2 = 6

4 Now add 2 to its neighbour 3 as 2 + 3 = 5
5 Now add 3 to its neighbour 2 as 3 + 2 = 5

6 Now add 2 to its neighbour 5 as 2 + 5 = 7

7 Place left most digit as it is  = 5

8 Write the digits so obtained ( blue coloured) from top to bottom as right to left

So Answer will be 5,75,564

How to  Multiply   543423 × 11= ? and many more  using  short cut Method



How to Multiply  two numbers   98 and  96 nearer to 100 

98 × 96 = (98 - 4)(2×4) = (94)(08) = 9408 (Only in mind )
Step 1 

Consider both the numbers as Num 1 = 98 and Num2 = 96.

Subtract Num 1 from 100 and write its Result 1 as one place .

Also Subtract Num 2 from 100 and write its as Result 2 second place .


Here Result 1 = 100 - 98 = 2
Result 2 = 100 - 96 = 4
Now multiply the results so obtained and mark it as Stepresult1.Stepresult1 = 4*2 = 8 = 08


Shortest method to multiply two  numbers





Step 2 


Subtract the result of  Result 1 (blue Answerfrom Num 2 i. e.(96 ).
Here we have
Stepresult2 = 96-2= 94

Final Answer = (Ist two digits are Stepresult2)(2nd two digits are Stepresult1)

= 9408

The result so obtained i. e. 9408 is the answer of product of two numbers
Shortest method to multiply two  numbers   in 2 seconds


More  Examples like  92 × 91 = ?



 How to Differentiate f(x) = (cos x )sin x


Consider    h(x) = (log cos x) ×(sin x)
then it derivative will be 

 f '(x) = f(x) Diff  (h(x))

⇒ f '(x) = f(x) [(log cos x) . Diff (sin x) + sin x Diff (log cos x)]

Therefore  f '(x) = f(x) [(log cos x) . cos x + sin x (-sin x ) /cos x)]


Therefore  f '(x) =  (cos x )sin x [(log cos x) . cos x - sin x .tan x]



Differentiate w.r.t. 'x' f(x) cos x sin x + (sin x) x 





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 

  [  -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 , Types of matrix  ,

Equality of Matrices And  Determinent 



Condition 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 × 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 do not match .


How to Multiply Matrices 



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   , 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)


How to solve system of linear equations







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



 x  - y + z   = 4
2x + y - 3z  = 0
 x + y + z    = 2
---------------------------------------------------------------------------------
How to multiply Two Matrices 



Matrix Multiplication





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


How to multiply 3×3 matrix with 3×3 matrix

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



multiplication of matrices


HOW TO MULTIPLY TWO MATRICES || PRODUCT OF TWO MATRICES

Solving the Quadratic Equation



1 Factorisation Method

2 Completing the Square


3 Quadratic Formula

Completing the Square



1 Shift the constant term to right hand side of equal sign.

2 Complete the square in left side and add the term which is missing and adjust the added term on the right side.

3 Equate the Left hand term to Right hand term.

Let us consider a Quadratic Equation

9x2 – 15x + 6 = 0
To make the complete square add the missing term   (b)2  and  subtract the same term





 (3x)2  – 2*3x *(5/2)+ (5/2)2–  (5/2)2 +6 = 0

 {3x-(5/2)}2 -(25/4)+6 = 0

{3x - (5/2)}2 -(1/4) = 0

{3x-(5/2)}2  = (1/4)                      Taking square roots
3x - (5/2) = (1/2)  or  3x-(5/2) = - (1/2)  
3x = (1/2)+(5/2)  or 3x = -(1/2) + (5/2)
3x = 6/2 or 3x = 4/2
x = 1 or    3x = 2
x = 1 or    x = 2/3

The roots of the given equation are 1 and   2/3





 Learn how to apply Factorisation Method    and      Quadratic Formula



Conclusion



In this post I have discussed some of my post published earlier in chronological order  . If this post helped you little bit, then please share it with your friends to benefit them, comment your views on it and also like this post to boost me and to do better, and also follow me on my Blog .We shell meet in next post till then Bye .
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