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Hyperbolic Functions V.S.A.Q.’S

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These solutions designed by the ‘Basics in Maths‘ team. These notes to do help the intermediate First-year Maths students.

Inter Maths – 1A two marks questions and solutions are very useful in IPE examinations.


Hyperbolic Functions

Question 1

Prove that for any x∈ R, sinh (3x) = 3 sinh x + 4 sinh3 x

Sol:

sinh (3x) = sinh (2x + x)

                  = sinh 2x cosh x + cosh 2x sinh x

                 = (2 sinh x cosh x) cosh x + (1 + 2 sinh2 x) sinh x

                 = 2sinh x cosh2 x + sinh x + 2 sinh3 x

                 = 2 sinh x (1 + sinh2 x) + sinh x + 2 sinh3 x

                 = 2 sinh x + 2 sinh3 x+ sinh x + 2 sinh3 x

                 = 3 sinh x + 4 sinh3 x

Question 2

If cosh x =Hyperbolic Functions 2 , find the values of (i) cosh 2x and (ii) sinh 2x

Sol:

Given cosh x =Hyperbolic Functions 2

 Cosh 2x = 2 cosh2 x – 1

                 = 2. Hyperbolic Functions 2 – 1

                 =Hyperbolic Functions 3

Sinh2 2x = cosh2 2x – 1

                 = Hyperbolic Functions 4– 1

                 = Hyperbolic Functions 5 – 1

       =Hyperbolic Functions 6

  Sinh2 2x   = Hyperbolic Functions 7

Question 3

If cosh x = sec θ then prove that tanh2Hyperbolic Functions 8= tan2Hyperbolic Functions 9

Sol:

tanh2  =Hyperbolic Functions 10

              = Hyperbolic Functions 11

             =Hyperbolic Functions 12

             =Hyperbolic Functions 13

             = tan2Hyperbolic Functions 9

Question 4

If sinh x = 5, then show that x =Hyperbolic Functions 14

Sol:

Given, sinh x = 5

      ⟹ x = sinh-15    

We know that sinh-1x = Hyperbolic Functions 15 

        ⟹ x =  Hyperbolic Functions 16     

               x  =  Hyperbolic Functions 14

Question 5

Show that tanh-1 = Hyperbolic Functions 17  log3

Sol:

Given tanh-1

We know that tanh-1 x = Hyperbolic Functions 17 Hyperbolic Functions 18   

 tanh-1  = Hyperbolic Functions 17 Hyperbolic Functions 19  

                 = Hyperbolic Functions 17Hyperbolic Functions 20   

                 = Hyperbolic Functions 17  log3

Question 6

For x, y ∈ R prove that sinh (x + y) = sinh (x) cosh (y) + cosh (x) sinh (y)

Sol:

R.H.S = sinh (x) cosh (y) + cosh (x) sinh (y)

Hyperbolic Functions 21

= Hyperbolic Functions 22 

= sinh (x + y)

Question 7

For any x∈ R, prove that cosh4 x – sinh4 x = cosh 2x

Sol:

cosh4 x – sinh4 x = (cosh2 x)2 – (sinh2 x)2

                                 = (cosh2 x + sinh2 x) (cosh2 x – sinh2 x)

                                 = 1. cosh 2x

                                 = cosh 2x

Question 8

Prove thatHyperbolic Functions 24 = cosh x + sinh x

Sol:

Hyperbolic Functions 23 

= cosh x + sinh x

Question 9

If sin hx = ¾ find cosh 2x and sinh 2x.

Sol:

Given sin hx = ¾

We know that cosh2 x = 1 + sinh2 x

                                           = 1 + (3/4)2

                                           = 1 + 9/16

                                           = 25/16

cos hx = 5/4

cosh 2x = 2cosh2 x – 1

                = 2(25/16) – 1

                = 25/8 – 1

                = 17/8

Sinh 2x = 2 sinh x cosh x

                = 2 (3/4) (5/4)

                = 15/8

Question 10

Prove that (cosh x – sinh x) n = cosh nx – sinh nx

Sol:

  Hyperbolic Functions 25

Hyperbolic Functions 26

∴ (cosh x – sinh x) n = cosh nx – sinh nx

 


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