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 {2 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 . {2 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 - sin² 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      


Hence the Proof

Prove that cos 4x = 8 cos⁴ x - 8 cos² x + 1

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 = 2(2cos² x -1)² -1

It is the square of 2cos² x -1

cos 4x = 2(2cos² x -1)² -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 = Sin (2x+x)

Sin 3x = sin 2x cos x + cos 2x sin 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 = 1- sin² 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 can prove that cos 3x= 4 cos³ x - 3 cos x

For learning and memorising more trigonometric formulas

visit here for  EASY TRIGONOMETRY PART 1

and EASY TRIGONOMETRY PART 2

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

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

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WHAT ARE LINEAR INEQUALITIES AND HOW TO SOLVE INEQUALITIES GRAPHICALLY

   Today  we  are  going  to  discuss  linear   inequalities ,    solving inequalities , linear inequations of one variable , linear  inequalities in two variables , graphing inequalities , solving linear inequalities , system    of   linear   inequalities ,  linear     inequalities  examples , application  of  linear  inequations and its   formation with the help of some examples.

Linear Inequation

An Inequation is a statement which involve the sign of  Inequality  < , > , ≤   ,  ≥  ,  ≠  etc

An equation which contains variables of 1st degree and does not  contains product of variables  is called Linear Inequation .

3x + 5 < 2  , 6x - 5 > 7 , 1 - 3x  > 10 , 2x - 3 ≤ 5

These are called Linear Inquations  Of One Variable.

3x - 5y  ≤  15 , 4x+3y  ≥ 12 ,

6x - 4y < 24  , 3x + 5y  ≥ 15,

2x - 5y  ≤  10,  3x + 7y  ≥ 21,

These are called Linear Equations  Of two Variables .

Solve the Inequation  1 - 2x  < - 8

1 - 2x  <  - 8

Shifting the constant term to R.H.S.

⇒  - 2x  <  - 8-1

 ⇒  - 2x  <  - 9

Isolating x from L.H.S by dividing  with -2 on both sides  and change the sign of Inequality.

Note : Whenever we divide or Multiply any Inequality with minus number ,then the sign of that inequality  changes , i.e.  <  to  > and  < to  >  or ≤  to   ≥  and   ≤    to   ≥ .

 ⇒  - 2x/-2  >  - 9/-2

⇒ x  >  9/2

It means every values greater than  9/2  will satisfy the given Inequality

Solve   the Inequalities   -4x + 1 ≥  0  ,  3 - 4x < 0

Consider 1st Inequality

- 4x + 1 ≥ 0  Shifting 1 to RHS

⇒ - 4x  ≥ 0 -1  Dividing with -4 on both sides to isolate x

⇒ - 4x /-4   ≥  - 1/ -4  ,change the sign of Inequality

 ⇒ x  ≤   1/4  = 0.25  -------(1)

Consider 2nd  Inequality

 ⇒    3 - 4x  <  0  Shifting 3 to RHS

 ⇒   - 4x  <  -3 Dividing with -4 on both sides to isolate x

⇒     -4x/-4  <  -3/-4          , change the sign of Inequality

⇒    x  >  3/4  = 0.75 ---------(2)

Now combine  the (1) and (2) ,we get NO Solution , Because simultaneously x can not be greater than 3/4 and less than 1/4

Solve -12 ≤  4 -3x -5 < 2

 Given   - 12  ≤  4 -3x -5 < 2 , 

To solve such types of Inequality Multiply  throughout   by -5 ( i. e. with denominator ) , if the multiplicand is negative then change the sign of all the inequalities

⇒   -12×(-5)   ≥  4 -3x -5 × (-5) > 2 × (-5) , 

 ⇒     60   ≥ 4 - 3x  > -10 , 

Add or Subtract the -ve of constant number  appearing with x (Here -ve of 4 ) to eliminate the constant term in middle of the inequality .

 ⇒     60 - 4   ≥ 4 - 3x - 4 > -10 - 4

 ⇒     56   ≥  - 3x  > -14  , 

Divide with co-eff of x to isolate  "x" i. e with -3

 ⇒     56/-3   ≤   - 3x/-3  < -14/-3

and change the sign of inequality if divisor is negative

⇒     -56/3   ≤  x < 14/3

⇒   All those values which are between  -563    and   143     are solution of given Inequality .

This video will help in understanding Linear Inequalities

 Solve   2x -3 4 + 8  ≥ 2+ 4x 3  and represent the Solution in number line

 Given 

2x - 3 4   + 8   ≥  2 +  4x 3  

To find the solution of such inequality where different denominators have two different number , we just multiply with the product of these two numbers which are in the denominators (Here 4 × 3  = 12 ) to each term of the inequality

 ⇒  2x - 3 4 × 12 + 8 ×12  ≥  2 × 12  +  4x 3 × 12 

⇒ (2x - 3) × 3 + 96  ≥  24  + ( 4x ) × 4 

⇒  6x - 9 + 96  ≥  24  +  16x  

⇒  6x + 87  ≥  24  +  16x  

⇒  6x -  16x  ≥  24  - 87 

⇒  - 10x  ≥  - 63   , Change the inequality sign after multiplication / division  with -ve number

⇒  x  ≤   63/10

Solve  - 3  ≤  4-7x2    ≤  18

Given - 3  ≤  4-7x2    ≤  18 ,

To Eliminate 2 from denominator  Multiply every term  with 2 

  ⇒   - 3 × 2  ≤  4-7x2   × 2  ≤  18 × 2

⇒   -6  ≤  4 -7x    ≤  36

⇒   -6 - 4 ≤  - 4 + 4 -7x  ≤  - 4+ 36

 ⇒   -10  ≤  -7x  ≤   32

 ⇒   -10/-7    ≥  -7x/-7   ≥  32/-7 , change the sign of inequality

⇒   10/7    ≥  x   ≥  -32/7

Hence set of all those numbers lying between 10/7 and -32/7 is the solution of the given inequality

Draw  the Diagram of the solution set of the constraints :   x  ≥ 0 ; y ≥ 0 , 4x + 7y ≤  28

In order to solve such a constraints , 1st of all draw graph of linear Equations x = 0 (which is known as y  axis) , y = 0 ( which is known as x-axis ) , 4x + 7y = 28 simultaneously .

As we know that graph of the  line x = 0 is the line y - axis and graph of the line  y = 0  is the line x - axis .

Now to draw the graph of line 4x + 7y = 28 , Put x = 0  and y = 0 in given line respectively and find the values of y and x . So if   x = 0 in the equation , we get y = 4 , and when we put y = 0 in line 4x + 7y = 28 , we get x = 7 .

Therefore       If    x  =  0  then  y = 4
and                 If    x =  7   then  y = 0 
we get two points A(0,4) and B(7,0) .

Now check the feasible region of each lines , as x  ≥ 0 implies right half of the Cartesian plan ( including y - axis ) and y  ≥ 0  implies upper half of the  Cartesian plan  ( including x - axis ) . So from these two inequalities we get the 1st quadrant as the common/feasible region .

Now  plots both the points A(0,4) and B(7,0) in the plan and draw a line passing through these two points. 

Now put (0,0) point in the inequality 4x + 7y ≤ 28 , If it comes out true then feasible region will be toward origin and if it comes out false then feasible region will be away from  origin .

Now mark  the common region from all the inequalities and  shade it , The shaded region will be the required/feasible region after graphing linear inequalities or system of inequalities . i.e The solution of  the given constraint.

Also  Read  HOW TO SOLVE HARD AND IMPOSSIBLE PUZZLES PART 3

Solve Graphically the System of Linear constraint  :   3x + 4y   ≥  12 ; 4x + 7y ≤ 28 ; x ≥ 0 ; y ≥  0

As we know from previous problem  x ≥ 0 ; y ≥  0 the common region from these two inequalities is in 1st quadrant.

Now Draw the graphs for  two  lines 3x + 4y = 12 and  4x + 7y =  28 .

Put x = 0 and y = 0 in 1st equation , we get  y = 3 and  x = 4 respectively . Therefore  two points will be  A(0,3) and B(4,0) .

Similarly put x = 0 and y = 0 in 2nd  equation respectively , we get  y = 4 and  x = 7 respectively .Therefore  two points will be  C(0,4) and D(7,0) .

Now  plots both the points A(0,3) and B(4,0) in the plan and draw a line passing through these two points. Also plots both the points C(0,4) and D(7,0) in the plan and draw a line passing through these two points. 

Now put (0,0) point in the inequality 3x + 4y   ≥  12 and 4x + 7y ≤ 28 , If it comes out true then feasible region of that inequality will be toward origin and if it comes out false then feasible region of that inequality  will be away from  origin .

 

At last mark  common region from all the inequalities x ≥ 0 ; y ≥  0 ; 3x + 4y   ≥  12 and 4x + 7y ≤ 28 and  shade it , after  graphing linear inequalities. The shaded region will be the required/feasible region ( As shaded in the above figure ) i.e. This is the  solution of the given constraint.

Solve Graphically the System of Linear constraint   2x+3y   ≥  6 ; x - 2y ≤ 2 ; 6x + 4y ≤  24 ; -3x +2y ≤ 3   x≥ 0 ; y ≥  0

As  from previous problems  x ≥ 0 ; y ≥  0 the common region from these two inequalities is in 1st quadrant.

Now Draw the graphs for  two linear lines 2x + 3y   = 6   ; 6x + 4y = 24 and  -3x +2y = 3 .

Put x = 0 and y = 0 in 1st equation 2x + 3y  =  6 , we get  y = 2 and  x = 3 respectively . Therefore  two points will be  A(0,2) and B(3,0) .

Now  put x = 0 and y = 0 in 2nd  equation x - 2y = 2, we get  y = -1 and  x = 2 respectively .Therefore  two points will be  C(0,-1) and D(2,0) .

Put x = 0 and y = 0 in 3rd equation 6x + 4y =  24 , we get  y = 6 and  x = 4 respectively . Therefore  two points will be  E(0,6) and F(4,0) .


Also Put x = 0 and y = 0 in 1st equation -3x + 2y = 3, we get  y = 3/2 and  x = -1 respectively . Therefore  two points will be  G(0,3/2) and H(-1,0) .

Now  plots both the points A(0,2) and B(3,0) in the plan and draw a line passing through these two points. Also plots both the points C(0,-1) and  D(2,0)  in the plan and draw a line passing through these two points. 

Also plots both the points E(0,6) ,F(4,0)  and G(0,3/2) and H(-1,0)
in the plan and draw a line passing through these two points. therefore there will be four lines in 1st quadrant.

Now put (0,0) point in the inequality 2x + 3y   ≥  6 ;  x - 2y  ≤  2 ; 6x + 4y ≤  24 ; -3x + 2y ≤ 3 , If it comes out true then feasible region of that inequality will be toward origin and if it comes out false then feasible region of that inequality  will be away from  origin .

At last mark  common region from all the inequalities x ≥ 0 ; y ≥  0 ; 3x + 4y   ≥  12 and 4x + 7y ≤ 28 and  shade it , The shaded region will be the required/feasible region. ( As shown in the figure above  ) i.e. This is the  solution of the given constraint.

Also Read  : Sets , Types of Set , Union ,Intersection , Complement of  set  and venn Diagrams

Application of Linear Inequalities

Problem :   In the  first four  papers each of 100 marks , Ravi got 99 , 88 , 77 , 96 marks . If he wants an average of  ≥ 80 marks and ≤  85 marks , Find  Find the range of Marks he should score in the fifth paper . 

Let Marks secured by Ravi in Fifth Paper  = x

Then as we know that average is the sum of  the marks secured in 

 five papers and divided by 5. 

Then according to given conditions the average must be grater than 80  and less than 85. Then Mathematically it can  be translated in linear equation  as follows  

80  ≤  99+88+77+96 + x 5  ≤  85

80  ≤  360 + x 5  ≤  85 , 
Now Multiplying with 5 to eliminate denominator 

80 × 5 ≤  360 + x 5 × 5 ≤  85 × 5 , 

400 ≤ 360 + x ≤  425 , 

Subtracting 360 throughout the inequation

400 - 360 ≤ 360 + x -360 ≤  425 - 360 , 

40 ≤ x ≤  65

• It means in order to get average of  greater than equal to  75  and  less than equal to   85  marks , he should score in the range of  40 to 65 marks in the Fifth Paper.

Problem : The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side . If the perimeter of the triangle is more than  166 cm then find the minimum length of the shortest side. 

Let the shortest side of the triangle  = x cm
It is given that longest side is twice the shortest side 
∴ Longest side of triangle  = 2x cm,

given that third  side is 2 cm longer than shortest side.
∴ Third   side of triangle =  x + 2  cm 

Then translating these line to mathematical form ( perimeter of triangle is more than 166 ) , we get

( x  ) + ( 2x ) + ( x + 2) > 166

 ⇒   4x + 2  > 166 

⇒   4x   > 166 - 2 

⇒   4x  > 164

⇒   x  >  166 /4

⇒   x  >  41

This implies the shortest side of the triangle must be 41 cm in order to satisfies  all the conditions .

Perimeter : Perimeter of a triangle  is the sum of all the sides of  given triangle .

Conclusion

In this post I have discussed Linear Inequations , linear inequations of one variable , solving linear inequalities, graphing linear inequalities,system of inequalities, linear inequalities in two variables, system of linear inequalities, linear inequalities examples,  and Application of linear inequations . 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 .

Let us learn Multiplication , division , arithmatic and simplification short cut ,tips and  tricks after buying this Book of Magical Mathematics .

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

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