Showing posts with label cbse. Show all posts
Showing posts with label cbse. Show all posts

## HOW TO PROVE TRIGONOMETRIC IDENTITIES || TRIGONOMETRY

Proof of trigonometric identities , trigonometric identities problems, proving trigonometric identities formulas,these trigonometric identities of class 10, fundamental trigonometric identities,trigonometric identities class 11 and its formation with the help of some examples.

## How to prove Identity

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

## Proof

1st of all  rewrite 3x as 3.2x

L.H.S. = cos 6x =  cos (3.2x)

Now using the result cos 3θ = 4cos3 θ - 3 cos θ  -----(1)

Replacing θ as 2x in (1), we get

L.H.S. = 4cos3 2x - 3 cos 2x  -----------(2)

Now using the result  1+ cos 2θ = 2 cos2 θ

⇒ cos 2θ = 2 cos2 θ -1

Replacing cos 2x = 2 cos2 x -1 in (2), we get

L.H.S.= 4 {2cos2 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}

Taking the product of powers to simplify it

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

Multiply by 4 in 1st term and multiply by -3 in 2nd term

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

Adding the like powers terms and arranging in descending order

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

Hence the Proof

## 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
Hence the Proof

## 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

Hence the Proof

## 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

cos 2x = cos2 x - sin2 x

Hence the Proof

Using the result
1+cos 2θ = 2cos2 θ
cos 2θ = 2cos2 θ -1 -------------(1)
Replacing θ with 2x in eq (1)
1+ cos 4x = 2cos2 2x
cos 4x = 2cos2 2x -1

Again using  cos 2θ = 2cos2 θ -1

cos 4x = 2Sq(2cos2 x -1) -1

It is the square of 2cos2 x -1

cos 4x = 2Sq(2cos2 x -1) -1

cos 4x = 2(4cos4 x +1 - 4cos2 x) -1

cos 4x = 8cos4 x +2 - 8cos2 x -1

cos 4x =  8cos4 x - 8cos2 x +1

Hence the Proof

## What is the value of sin3x?

To find the value of sin 3x ,  use this formula which contain sin (A+B)
therefore sin (A+B) = sin A cos B cos A sin B——-(1)
put A = 2x and B = x in (1)
then Sin 3x = sin 2x cos x + cos 2x sin x

## As we know that cos 2x = 1 - 2sin3 x and sin 2x = 2 sin x cos x

sin 3x = (2 sin x cos x) cos x + (1 - 2sin3 x ) sin x
sin 3 x = 2 sin x cos2 x + sin x -  2sin3 x

As we know that cos2 x = 1sin2 x

sin 3x= 2 sin x (1-sin3 x) + sin x - 2sin3 x
sin 3x = 2 sin x -2 sin3 x + sin x - 2sin3 x
sin 3x = 3 sin x - 4 sin3 x

Similarly we prove that cos 3x= 4 cos3 x - 3 cos x
For learning and memorising more trigonometric formulas

## Conclusion

In this post I have discussed trigonometric identities ,trigonometric identities problems, proving trigonometric identities formulas . If this post helped you little bit, then please share it with your friends to benefit them, comment your views on it 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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## Classes 10th and 12th Question Papers Style

There is a big  news coming in  media regarding some changes are being planned by CBSE in the pattern of Question Papers of their board classes 10th and 12th .  The central Board of School Education ( CBSE ) has decided to introduce new pattern of  Question Papers for 10th and 12th classes as a part of revamp that would change examination  schedule for vocational  subjects , It would also be  implementing to other mains subjects.

## What is the need of Change in existing Pattern

According to the sources from the The Ministry of Human Resources and Development and some agencies , the new pattern of question papers of classes  of 10th and 12th will  not support the students who are dependent on  rote learning , it will be design to discourages such type of students .

This pattern  would also stop students from total  blind copying of text books at home and pasting or vomiting of texts in the  examination hall in the Answer Book .

The new pattern of question Papers of these classes would test students on their analytical skill and reasoning abilities .

The CBSE also thinks that  its new steps  will also increase quality Education and better result of its institutions .

## These are the majors changes expected in new pattern of CBSE

The new question papers pattern  will be design to check the analytical thinking and art of problems solving of the students in the examination.

Annual Board exams to be completed before the 15th  of the March  and vocational  course Exams to be completed by the end of  February Every year .

More short answers type questions from 1 to 5 marks shall be included in the new pattern of the question paper . More emphasis would  be  on probing the critical thinking and ability of the students.

CBSE has already submitted its proposal to reform in examination pattern to the concered Ministry for approval .
The proposal for examination reforms is still in discussion stage and the CBSE has started working to implement new pattern by March 2020.

## Benefits  of proposed  Paper pattern

From the  Early completion of the Board  examination ,the examiners would have more time to evaluate  answer books of the students , which will result in early declaration  of Annual Board exams.
Renewed paper pattern will focus on simplifies and shorten the rules of affiliations and renewal for the school.

## At Last but not the least

I would like to thanks for devoting your valuable time for this post "Regarding proposal of CBSE to introduce new pattern of Question Paper " of my blog . If you liked this  blog/post, Do Follow me on my blog and share this post with your friends . We shall meet again in next post with solutions of most interesting mathematics topics and mind blowing puzzles ,till then Good Bye.

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## HOW TO LEARN INTEGRATION FORMULAE/FORMULAS USING TRICKS

Let us learn and remember most Important formulas of Integration , tips and tricks to learn algebraic ,most important differentiation questions for plus 2 maths, indefinite integration tricks and shortcuts trigonometric and by parts formulas in an easy and short cut manners.

## 1  ∫ sin x dx           =  - cos x +c

where "c" is called constant of Integration.

The integration of sin x is  - cos x ,then divide it with the derivative of its angle.

If we have to find the integration of  sin 2x , then we shall find it as

Step1    1st find the integration of sin x which is - cos x .

Step2    Divide it with the derivative of 2x ,which is 2, so

∫ sin 2x dx     =  - ( cos 2x) 2 + c ,
∫ sin 8x dx      =  - ( cos 8x) 8 + c ,

∫ sin  3x4  dx  =   - ( cos  3x4 )   3 4  + c ,
Therefore   ∫ sin nx dx =  - ( cos nx) n +c ,

## 6 ∫ cosec x dx = - log |cosec x - cot x | +c

If we want to integrate cosec√x .Then 1st of all we apply the formula of integration of cosec (any angle) then formula of integration of √x,So we have

∫  cosec√x dx = - (log | cosec√x -co√x | )(2√x) +c

## 7  ∫ sec2 x dx = tan x + c

Because the derivative of tan x is sec 2 x , So the Antiderivative or Integration of sec 2 x  is tan x .

∫ sec 2 √x dx     =  (2√x ) tan √x  + c

∫ cosec 2 dx = - cot x +c

Because the derivative of  cot x is  - cosec 2 x , So the  Anti derivative or Integration of  cosec 2 x is - cot x .

## 8 ∫ sec x tan x dx = sec x +c

Because the derivative of sec x is sec x tan x ,Therefore the integration of tan x sec x is sec
x .

If we want to integrate sec√x .tan √x .Then its  integration  will be sec √x,

sec √x tan √x   dx      =  √x   sec √x + c

## 9 ∫ cosec x cot x dx = - cosec x +c

Because the derivative of cosec x is    - cosec x cot x , Therefore the integration of tan x sec x is sec x .

∫ cosec √x cot √x dx = - ( 2 √x  cosec √x ) +c

## Conclusion

Thanks for devoting your valuable time for this post "HOW TO LEARN INTEGRATION  FORMULAE / FORMULAS VERY   EASILY" of my blog . If you liked this  blog/post, Do Follow me on my blog and share this post with your friends . We shall meet again with new post  ,till then Good Bye.

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