Trigonometric Functions Vs Hyperbolic Functions

Trigonometric Functions vs Hyperbolic Functions Definitions ix

-ix

e x - e- x 2 x e + e- x cosh x = 2 sinh x tanh x = cosh x 1 csch x = sinh x 1 sech x = cosh x cosh x coth x = sinh x Inverse Hyperbolic Functions Expressed in Terms of Logarithmic Functions

e -e 2i ix e + e-ix cos x = 2 sin x tan x = cos x 1 csc x = sin x 1 sec x = cos x cos x cot x = sin x

sin x =

sinh x =

(

)

sinh-1 x = ln x + x 2 + 1 -¥ < x < ¥

(

)

cosh-1 x = ln x + x 2 -1

x ³ 1 ( cosh -1 x > 0 is principal value) 1 æ1 + x ö÷ tanh -1 x = ln çç ÷ 2 çè1- x ÷ø -1 < x < 1 æ1 ö 1 csch -1 x = ln ççç + 2 + 1÷÷÷ ÷ø x èç x x¹0 æ1 ö 1 sech -1 x = ln ççç + 2 -1÷÷÷ ÷ø çè x x 0 < x £ 1 ( sech -1 x > 0 is principal value) 1 æ x + 1ö÷ coth -1 x = ln çç ÷ 2 çè x -1÷ø x > 1 or x < -1 Relationship between Hyperbolic and Trigonometric Functions sin (ix ) = i sinh x sinh (ix ) = i sin x cos (ix) = cosh x

cosh (ix ) = cos x

tan (ix ) = i tanh x

tanh (ix ) = i tan x

csc (ix ) = -i csch x

csch (ix ) = -i csc x 1

sec (ix ) = sech x

sech (ix ) = sec x

cot (ix ) = -i coth x

coth (ix ) = -i cot x

Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions sin -1 (ix ) = i sinh -1 x sinh -1 (ix ) = i sin -1 x

cos-1 x = i cosh-1 x tan -1 (ix ) = i tanh-1 x

cosh-1 x = i cos-1 x tanh -1 (ix ) = i tan-1 x

csc-1 (ix ) = -i csch -1 x

csch -1 (ix) = -i csc-1 x

sec-1 x = i sech-1 x cot -1 (ix ) = -i coth -1 x

sech -1 x = i sec-1 x coth -1 (ix ) = -i cot -1 x

Relationships among Trigonometric Relationships among Hyperbolic Functitons Functitons 2 2 cos x + sin x = 1 cosh 2 x - sinh 2 x = 1 sec 2 x - tan 2 x = 1 sech 2 x + tanh 2 x = 1 csc2 x - cot 2 x = 1 coth 2 x - csch 2 x = 1 Relations between Inverse Relations between Inverse Hyperbolic Trigonometric Functions Functions 1 1 csc-1 x = sin -1 csch-1 x = sinh -1 x x 1 1 sec-1 x = cos-1 sech -1 x = cosh -1 x x 1 1 cot -1 x = tan -1 coth -1 x = tanh -1 x x Negative Angle Formulas sin (-x ) = - sin x sinh (-x ) = - sinh x cos (-x ) = cos x

cosh (-x ) = cosh x

tan (-x ) = - tan x

tanh (-x ) = - tanh x

csc (-x ) = - csc x

csch (-x ) = - csch x

sec (-x) = sec x

sech (-x ) = sech x

cot (-x) = - cot x sin (-x ) = - sin -1

coth (-x) = - coth x -1

sinh -1 (-x ) = - sinh -1 x

x

cos-1 (-x ) = p - cos-1 x

cosh -1 (-x) = pi - cosh -1 x

tan -1 (-x ) = - tan -1 x

tanh -1 (-x ) = - tanh -1 x

cot -1 (-x) = p - cot -1 x

coth -1 (-x) = - coth -1 x

sec-1 (-x) = p - sec-1 x

sech -1 (-x) = pi - sech -1 x

csc-1 (-x ) = - csc-1 x

csch -1 (-x) = - csch -1 x

sin ( x  y )

Addition Formulas sinh ( x  y )

= sin x cos y  cos x sin y

= sinh x cosh y  cosh x sinh y 2

cos ( x  y )

cosh ( x  y )

= cos x cos y  sin x sin y

= cosh x cosh y  sinh x sinh y

tan ( x  y ) =

tanh ( x  y )

tan x  tan y 1  tan x tan y

=

cot ( x  y )

tanh x  tanh y 1  tanh x tanh y

coth ( x  y )

coth x coth y  1 coth y  coth x Half Angle Formulas x x 1- cos x cosh x -1 sinh =  sin =  2 2 2 2 (+ if x 2 is in quadrant I or II (+ if x > 0 =

cot x cot y  1 cot y  cot x

=

- if x 2 is in quadrant III or IV)

- if x < 0 )

x 1 + cos x cos =  2 2

cosh

x cosh x + 1 = 2 2

(+ if x 2 is in quadrant I or IV - if x 2 is in quadrant II or III)

x 1- cos x = 2 1 + cos x sin x 1- cos x = = 1 + cos x sin x = csc x - cot x (+ if x 2 is in quadrant I or III

x cosh x -1 = 2 cosh x + 1 sinh x cosh x -1 = = cosh x + 1 sinh x = coth x - csch x (+ if x > 0

tan

tanh

- if x 2 is in quadrant II or IV) - if x < 0 ) Multiple Angle Formulas sin 2 x = 2sin x cos x sinh 2 x = 2sinh x cosh x 2 2 cos 2 x = cos x - sin x cos 2 x = cosh 2 x + sinh 2 x

= 2 cos 2 x -1

= 2 cosh 2 x -1

= 1- 2sin 2 x 2 tan x tan 2 x = 1- tan 2 x sin 3 x = 3sin x - 4sin 3 x cos 3x = 4 cos3 x - 3cos x 3 tan x - tan 3 x tan 3x = 1- 3 tan 2 x sin 4 x = 4sin x cos x - 8sin 3 x cos x cos 4 x = 8cos 4 x - 8cos 2 x + 1 4 tan x - 4 tan 3 x tan 4 x = 1- 6 tan 2 x + tan 4 x

= 1 + 2sinh 2 x 2 tanh x tanh 2 x = 1 + tanh 2 x sinh 3 x = 3sinh x + 4sinh 3 x cos 3x = 4 cosh 3 x - 3cosh x 3 tanh x + tanh 3 x tanh 3 x = 1 + 3 tanh 2 x sinh 4 x = 8sinh 3 x cosh x + 4sinh x cosh x cosh 4 x = 8cosh 4 x - 8cosh 2 x + 1 4 tanh x + 4 tanh 3 x tanh 4 x = 1 + 6 tanh 2 x + tanh 4 x 3

Powers of Trigonometric Function 1 1 sin 2 x = - cos 2 x 2 2 1 1 cos 2 x = + cos 2 x 2 2 3 1 sin 3 x = sin x - sin 3 x 4 4 3 1 cos3 x = cos x + cos 3x 4 4 3 1 1 sin 4 x = - cos 2 x + cos 4 x 8 2 8 3 1 1 cos 4 x = + cos 2 x + cos 4 x 8 2 8 Sum, Difference and Product of Trigonometric Functions sin x + sin y

x+ y x- y cos 2 2 sin x - sin y x+ y x- y sin = 2 cos 2 2 cos x + cos y x+ y x- y cos = 2 cos 2 2 cos x - cos y x+ y x- y = 2sin sin 2 2 sin x sin y 1 = - éë cos ( x + y ) - cos ( x - y )ùû 2 cos x cos y = 2sin

1 = éë cos ( x + y ) + cos ( x - y )ùû 2 sin x cos y 1 = éësin ( x + y ) + sin ( x - y )ùû 2 cos x sin y 1 = éësin ( x + y ) - sin ( x - y )ùû 2 d sin x = cos x dx d cos x = - sin x dx

Powers of Hyperbolic Function 1 1 sinh 2 x = cosh 2 x 2 2 1 1 cos 2 x = cosh 2 x + 2 2 1 3 sinh 3 x = sinh 3x - sinh x 4 4 1 3 cos3 x = cosh 3 x + cosh x 4 4 3 1 1 sinh 4 x = - cosh 2 x + cosh 4 x 8 2 8 3 1 1 cos 4 x = + cosh 2 x + cosh 4 x 8 2 8 Sum, Difference and Product of Hyperbolic Functions sinh x + sinh y

x+ y x- y cosh 2 2 sinh x - sinh y x+ y x- y sinh = 2 cosh 2 2 cosh x + cosh y x+ y x- y cosh = 2 cosh 2 2 cosh x - cosh y x+ y x- y sinh = 2sinh 2 2 sinh x sinh y 1 = éë cosh ( x + y ) - cosh ( x - y )ùû 2 cosh x cosh y 1 = éë cosh ( x + y ) + cosh ( x - y )ùû 2 sinh x cosh y 1 = éësinh ( x + y ) + sinh ( x - y )ùû 2 cosh x sinh y 1 = éësinh ( x + y ) - sinh ( x - y )ùû 2 Derivatives d sinh x = cosh x dx d cosh x = sinh x dx = 2sinh

4

d tan x = sec2 x dx d csc x = - csc x cot x dx d sec x = sec x tan x dx d cot x = - csc 2 x dx d 1 sin -1 x = dx 1- x 2

d tanh x = sech 2 x dx d csch x = - csch x coth x dx d sech x = - sech x tanh x dx d coth x = - csch 2 x dx 1 d sinh -1 x = 2 dx x +1

d 1 cos-1 x = dx 1- x 2

d 1 cosh -1 x =  2 dx x -1 -1 (+ if cosh x > 0 , x > 1 - if cosh -1 x < 0 , x > 1 )

d 1 csc-1 x = dx x x 2 -1

d 1 tanh-1 x = dx 1- x 2 ( -1 < x < 1 ) 1 d csch -1 x = dx x 1+ x2

d 1 sec-1 x = dx x x 2 -1

d 1 sech -1 x = dx x 1- x 2

d 1 tan -1 x = dx 1+ x2

d 1 cot -1 x = dx 1+ x2

d 1 coth -1 x = dx 1- x 2 ( x > 1 or x < -1 ) Integrals

ò sin x dx = - cos x ò cos x dx = sin x ò tan x dx = - ln cos x ò csc x dx = ln (csc x - cot x) = ln tan

ò sinh x dx = cosh x ò cosh x dx = sinh x ò tanh x dx = ln cosh x x

ò csch x dx = ln tanh 2

x 2

ò sec x dx = ln (sec x + tan x)

ò sech x dx = sin

æ x pö = ln tan çç + ÷÷÷ çè 2 4 ø

2

(tanh x)

= tan -1 (sinh x)

ò cot x dx = ln sin x

ò sin

-1

ò coth x dx = ln sinh x

x sin 2 x x dx = 2 4

ò sinh

5

2

x dx =

sinh 2 x x 4 2

ò cos

2

x dx =

x sin 2 x + 2 4

ò cosh

ò tan x dx = tan x - x ò csc x dx = - cot x ò sec x dx = tan x ò cot x dx = - cot x - x ò sec x tan x dx = sec x ò csc x cot x dx = - csc x

2

x dx =

sinh 2 x x + 4 2

ò tanh x dx = x - tanh x ò csch x dx = - coth x ò sech x dx = tanh x ò coth x dx = x - coth x ò sech x tanh x dx = - sech x ò csch x coth x dx = - csch x

2

2

2

2

2

2

2

2

6