Tiêu đề XI - maths - chapter 11 - HYPERBOLIC FUNCTIONS (156-165).pdf ✅

HYPERBOLIC FUNCTIONS JEE MAINS - VOL - II 156  If x is any real number then 2 3 1 ... .... 1 2 3 n x x x x x e n         is called the exponential series .    2 3 1 ........... 1 .... 1 2 3 n n x x x x x e n           .  The function x e can be written as 2 2 x x x x x e e e e e       for all x;  (i) sinh 2 x x e e x    3 5 .... 1 3 5 x x x     , x (ii) cosh 2 x x e e x    2 4 1 ..... 2 4 x x     , x (iii) sinh tanh cosh x x x  , x (iv) 1 sec , cosh hx x  x (v) 1 cos sinh echx x  , x   0 (vi) cosh coth sinh x x x  , x   0  Hyperbolic functions are not circular functions and hence are not meant to use trigonometric identities.  Hyperbolic Identities: (i) 2 2 cosh sinh 1, x x x     (ii) 2 2 sech tanh 1, x x x     (iii)   2 2 coth cos h 1, 0 x ec x x       W.E-1:sinh 3/ 4 x  then cosh x  Sol: 2 2 2 2 cosh sinh 1 cosh 1 sinh x x x x      9 5 1 cosh 16 4     x  Properties of Hyperbolic Functions: (i) sin h( ) x = -sin h x (ii) cosh( ) x = cosh x (iii) tanh( ) x = -tanh x (iv) coth( ) x = -coth x (v) sech( ) x = sech x (vi) cosech( ) x = -cosech x  (i) sinh sinh cosh cosh sinh  x y x y x y     (ii) cosh cosh cosh sin h sinh  x y x y x y     (iii)   tanh tanh tanh 1 tanh tanh x y x y x y     (iv)   coth coth 1 coth coth coth x y x y y x      (i) 2 2 tanh sinh 2 2sinh cosh 1 tanh x x x x x    (ii) 2 2 2 2 1 tanh cosh 2 cosh sinh 1 tanh x x x x x      (iii) 2 2 tanh tanh 2 1 tanh x x x   (iv) 1 tanh sinh 2 cosh 2 1 tanh x x x x     W.E-2: tanh 3/ 5 x  then cosh 2 x  Sol: 2 2 1 tanh 34 17 cosh 2 1 tanh 16 8 x x x       (i) sinh3 x =3sinh x+4sinh3 x (ii) cosh3 x = 4cosh3 x - 3 cosh x (iii)   3 2 3 tanh tanh tanh 3 1 3 tanh x x x x    W.E-3: tanh 1 2 x  then tanh 3 x  Sol:         3 2 3tanh tanh 13 tanh 3 1 3tanh 14 x x x x     HYPERBOLIC FUNCTIONS SYNOPSIS
157 HYPERBOLIC FUNCTIONS JEE MAINS - VOL - II  2 2 sin h ( )sin h ( ) sin sin h x y x y h x y      2 2 cos h ( )cos h ( ) cos sin h x y x y h x y     (cos h sin h ) (cos h ( ) sin ( )) n nx x x nx h nx e     (cosh sinh ) (cosh( ) sin ( )) n nx x x nx h nx e      cosh 2nx + sinh 2nx = 1 tanh 1 tanh n x x          cosh x, sech x are even functions.  sinh x, cosech x, tanh x and coth x are odd functions. W.E.4: Show that f x x    cosh is an even function Sol:       cosh   2 2 x x x x e e e e f x x f x              None of the six hyperbolic functions are periodic.  sinh x, tanh x ,coth x and cosech x are one -one functions but cosh x and sech x are not one one functions as cosh cosh sec sec  x x and h x h x    for all x       sinhx + coshx, sinhx - coshx are bijective functions  Domain and range of hyperbolic functions: Function Domain Range Sinh x R R Cosh x R [1, )  Tanh x R 1,1 Coth x R - {0} R-[-1,1] Cosech x R - {0} R - {0} Sech x R (0,1]  Since hyperbolic functions are defined in terms of exponential functions. Therefore Inverse hyperbolic functions can be expressed in terms of logarithmic functions.  Inverse Hyperbolic Functions in Terms of Logarithmic Functions: i)   2 log 1 -1 e sinh x x x     x  W.E-5: If sinh 3 x  . Then x = .... Sol:     1 2 sinh 3 sinh 3 log 3 3 1 x x        = log 3 10    ii) cosh-1(x)=   2 log 1 1 e x x forx    W.E - 6: If cosh 3 x  then x = ..... Sol:       1 2 cosh 3 cosh 3 log 3 3 1 log 3 8 x x          iii) tanh-1(x) =   1 1 log 1,1 2 1 x for x x           Similarly we have iv) coth-1( x ) = 1 1 log 1 2 1 x for x x          v) sech-1( x ) 2 1 1 log for (0,1] e x x x           vi) cosech-1( x ) = 2 1 1 log for 0 e x x x          = 2 1 1 loge x x           for x < 0  Domain and Range of inverse Hyperbolic Functions: Function Domain Range Sinh-1 x R R Cosh-1 x [1,) [0,) Tanh-1 x (-1,1) R Coth-1 x R 1,1    R - {0} Sech-1 x (0,1] [0, )  Cosech-1 x R - {0} R - {0}  sinh-1( x ) =   1 2 cosh 1 x   1 1 cos ech x         1 2 tanh 1 x x         
HYPERBOLIC FUNCTIONS JEE MAINS - VOL - II 158      1 1 2 cosh sinh 1 x x     1 1 sec h x         2 1 1 tanh x x             Euler’s Formula : cos sin ix e x i x    x R cos sin ix e x i x     x R  Hyperbolic Functions Using Euler’s Formula : sinh (i x ) = i sin x cosh (i x ) = cos x tanh (i x ) = i tan x coth (i x ) = -i cot x cosech (i x ) = -i cosec x sech (i x ) = sec x  sin sinh ix i x  , cos cosh ix x  tan tanh ix i x  cot coth ix i x   ,sec sec ix hx  cos cosec ecix i h x    1 1 sinh sin x i ix     1 1 cosh cos x i x     1 1 tanh tan x i ix     1. The domain of cos ech x is 1)  ,  2)    ,0 0,    3) 0, 4) ,0  2 The range of coth x is 1)    , 1 1,  ,  2) 1,1 3)  ,  4)  , 1 3. cosh 0 =   1) 0 2) 1 3) e 4) not defined 4. cosh 2 +sinh 2 =     (EAM-2000) 1) 1 e 2) e 3) 2 1 e 4) 2 e 5. sinh 3 -cosh 3 =     1) 3 e 2) 3 e  3) 3 e 4) 3 e 6. cosh2+ sinh2 n = 1) cosh 2 sinh 2  n n     2) cosh 2 sinh 2  n n     3) cosh 2 n 4) sinh 2 n 7. If cosh x=sec then cos ec h x  1) sin 2) cos 3) tan 4) cot 8. If sinh x = 3 4 then sinh 2x =   1) 5 8 2) 15 8 3) 7 8 4) 17 8 9. If tanh x = 3 5 then sinh (2 x ) = 1) 15 8 2) 15 17 3) 18 17 4) 17 8 10.     4 4 cosh sinh x x  = 1) cosh x 2) cosh2 x 3) sinh x 4) sinh2 x 11. If cosh x = 5 4 then cosh 3 x = 1) 61 16 2) 63 16 3) 61 63 4) 65 16 12. 3 1 2 sinh 2        = 1) log 2 18 e    2) log 3 8 e    3)log 3 8 e    4) log 8 27 e    13.   1 cosh 2  = 1)log 2 3 e    2)log 2 5 e    3) log 2 5 e    4) log 2 2 e    14. 1 1 T anh 3        = 1)   1 log 2 2 e 2) 2log 2 e   LEVEL - I (C.W)
159 HYPERBOLIC FUNCTIONS JEE MAINS - VOL - II 3)   1 log 2 2  e 4) 2log 2 e   15. 1 1 2Tanh 2        = 1) 0 2) log 2e 3) log 3e 4)log 4e 16.   1 cosech 4   = 1) 17 1 log 4 e        2) log 17 1 e    3)log 17 1 e    4) 17 1 log 4 e        17. If     1 sinh log 5 26 e x    then x = 1) 1 2) 2 3) 3 4)5 18. If   1 1 tanh log 1 e y x y then y           (EAM-1998) 1) x 2) 4 x 3) 2 x 4) 3 x 19. If   1 1 tanh log , 1 1 e x x a x x            then a= (EAM-2010) 1) 1 2) 2 3) 1 2 4) 1 4 20. If     1 cosh log 3 2 2 e k then k     1)1 2) 2 3) 3 4) 4 21.   1 sinh cot e   = 1) cot +cosec   2)  cot cos   ec 3) secθ-tanθ 4) secθ+tanθ 22.   -1 sech sinθ = 1) log cos 2 e        2) sin 2 loge        3)loge cos  4) cot 2 loge        23. If       1 1 sinh 2 sinh 3 cosh xthen x      (EAM-2009) 1)   1 3 5 2 10 2  2)   1 3 5 2 10 2  3)   1 12 2 50 2  4)   1 12 2 50 2  24. If e π θ tan + 4 2 x=log             then coshx= 1) secθ 2) cosecθ 3) sinθ 4) cosθ 25. cot sinh 4 loge If x then x                  1) tan 2 2) cot 2 3) tan 2 4) cot 2 26. If cosh2 99 thentanh x x   1) 5 7 2 2) 7 5 2 3) 5 7 2 4) 7 5 2 27. sinhix = 1) isinx 2) sinix 3) -isinx 4) isin(ix) LEVEL - I (C.W)-KEY 01) 2 02)1 03) 2 04) 4 05)2 06)1 07)4 08) 2 09) 1 10) 2 11) 4 12) 2 13) 1 14) 1 15) 3 16)1 17) 4 18) 3 19) 3 20) 3 21) 1 22) 4 23) 3 24) 1 25) 3 26) 2 27) 1 LEVEL - I (C.W)-HINTS 1. The domain of cosechx is    ,0 0,    2. The range of cothx is     , 1 1,    3. Cosh 0 1    4.     2 Cosh 2 sinh 2   e 5. 3 3 3 3 3 2 e e e e e         6.       2 cosh 2 sinh 2 n e n n   7. cosh sec x   2 2 cosh sinh 1 x x   2 2   sinh tan x    cos cot echx 