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Showing posts sorted by relevance for query trigonometry. Sort by date 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



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 
Hence the Proof


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

Hence the Proof


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   

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




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

welcome to this post of learning trigonometric formulas  .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 .
how-to-memorise-A-B-and-C-D-formulae



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. 1st   and 2nd terms are   cosine and  3rd and 4th 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. 1st   and 2nd terms are   cosine and  3rd and 4th  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.

how-to-memorise-A-B-and-C-D-formulae

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.
how-to-memorise-A-B-and-C-D-formulae

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.

how-to-memorise-A-B-and-C-D-formulae 
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.

how-to-memorise-A-B-and-C-D-formulae

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-formulae


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

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 ,trigonometry formulas for class 11 ncert,trigonometric functions class 11 notes,trigonometry class 11 tricks,,trigonometry formulas list,. 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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