Hyperbolic Functions

Hyperbolic functions - sinh, cosh, tanh, coth, sech, csch

DEFINITION OF HYPERBOLIC FUNCTIONS

Hyperbolic sine of x = sinh x = (e^x - e^-x)/2
Hyperbolic cosine of x = cosh x = (e^x + e^-x)/2
Hyperbolic tangent of x = tanh x = (e^x - e^-x)/(e^x + e^-x)
Hyperbolic cotangent of x = coth x = (e^x + e^-x)/(e^x - e^-x)
Hyperbolic secant of x = sech x = 2/(e^x + e^-x)
Hyperbolic cosecant of x = csch x = 2/(e^x - e^-x)

RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS

tanh x = sinh x/cosh x
coth x = 1/tanh x = cosh x/sinh x
sech x = 1/cosh x
csch x = 1/sinh x
cosh²x - sinh²x = 1
sech²x + tanh²x = 1
coth²x - csch²x = 1

FUNCTIONS OF NEGATIVE ARGUMENTS

sinh(-x) = -sinh x
cosh(-x) = cosh x
tanh(-x) = -tanh x
csch(-x) = -csch x
sech(-x) = sech x
coth(-x) = -coth x

ADDITION FORMULAS

sinh (x ± y) = sinh x cosh y ± cosh x sinh y
cosh (x ± y) = cosh x cosh y ± sinh x sinh y
tanh(x ± y) = (tanh x ± tanh y)/(1 ± tanh x.tanh y)
coth(x ± y) = (coth x coth y ± l)/(coth y ± coth x)

DOUBLE ANGLE FORMULAS

sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 2 cosh²x — 1 = 1 + 2 sinh²x
tanh 2x = (2tanh x)/(1 + tanh²x)

HALF ANGLE FORMULAS

sinh x/2 = ± [+ if x > 0, - if x < 0]
cosh x/2 =
tanh x/2 = ± [+ if x > 0, - if x < 0]
= <(sinh x)/(cosh x — 1) = (cosh x + 1)/sinh x

MULTIPLE ANGLE FORMULAS

sinh 3x = 3 sinh x + 4 sinh³ x
cosh 3x = 4 cosh³ x — 3 cosh x
tanh 3x = (3 tanh x + tanh³ x)/(1 + 3 tanh²x)
sinh 4x = 8 sinh³ x cosh x + 4 sinh x cosh x
cosh 4x = 8 cosh⁴ x — 8 cosh² x + 1
tanh 4x = (4 tanh x + 4 tanh³ x)/(1 + 6 tanh² x + tanh⁴ x)

POWERS OF HYPERBOLIC FUNCTIONS

sinh² x = ½cosh 2x — ½
cosh² x = ½cosh 2x + ½
sinh³ x = ¼sinh 3x — ¾sinh x
cosh³ x = ¼\\ cosh 3x + ¾cosh x
sinh⁴ x = 3/8 - ½cosh 2x + 1/8cosh 4x
cosh⁴ x = 3/8 + ½cosh 2x + 1/8cosh 4x

SUM, DIFFERENCE AND PRODUCT OF HYPERBOLIC FUNCTIONS

sinh x + sinh y = 2 sinh ½(x + y) cosh ½(x - y)
sinh x - sinh y = 2 cosh ½(x + y) sinh ½(x - y)
cosh x + cosh y = 2 cosh ½(x + y) cosh ½(x - y)
cosh x - cosh y = 2 sinh ½(x + y) sinh ½(x — y)
sinh x sinh y =    ½(cosh (x + y) - cosh (x - y))
cosh x cosh y = ½(cosh (x + y) + cosh (x — y))
sinh x cosh y = ½(sinh (x + y) + sinh (x - y))

EXPRESSION OF HYPERBOLIC FUNCTIONS IN TERMS OF OTHERS

In the following we assume x > 0. If x < 0 use the appropriate sign as indicated by formulas in the section "Functions of Negative Arguments"

	sinh x = u	cosh x — u	tanh x = u	coth x = u	sech x = u	esch x = u
sinh x	u	u	u/	l/	/u	1/u
cosh x		u	1/	u/	1/u	/u
tanh x	u/	/u	u	1/u		1/
coth x	/u	u/	1/u	u	1/
sech x	1/	1/u		/u	u	u/
csch x	1/u	1/	/u		u/	u

GRAPHS OF HYPERBOLIC FUNCTIONS

y = sinh x

y = cosh x

y = tanh x

y = coth x

y = sech x

y = csch x

INVERSE HYPERBOLIC FUNCTIONS

If x = sinh y, then y = sinh^-1 a is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single-valued.
The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.
sinh^-1 x = ln (x +)     -∞ < x < ∞
cosh^-1 x = ln (x + )     x ≥ 1 [cosh^-1 x > 0 is principal value]
tanh^-1x = ½ln((1 + x)/(1 - x))     -1 < x < 1
coth^-1 x = ½ln((x + 1)/(x - 1))     x > 1 or x < -1
sech^-1 x = ln ( 1/x + )     0 < x ≤ 1  [sech^-1 a; > 0 is principal value]
csch^-1 x = ln(1/x + )     x ≠ 0

RELATIONS BETWEEN INVERSE HYPERBOLIC FUNCTIONS

csch^-1 x = sinh^-1 (1/x)
sech^-1 x = cosh^-1 (1/x)
coth^-1 x = tanh^-1 (1/x)
sinh^-1(-x) = -sinh^-1x
tanh^-1(-x) = -tanh^-1x
coth^-1 (-x) = -coth^-1x
csch^-1 (-x) = -csch^-1x

GRAPHS OF INVERSE HYPERBOLIC FUNCTIONS

y = sinh^-1x

y = cosh^-1x

y = tanh^-1x

y = coth^-1x

y = sech^-1x

y = csch^-1x

RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS

sin(ix) = i sinh x	cos(ix) = cosh x	tan(ix) = i tanh x
csc(ix) = -i csch x	sec(ix) = sech x	cot(ix) = -i coth x
sinh(ix) = i sin x	cosh(ix) = cos x	tanh(ix) = i tan x
csch(ix) = -i csc x	sech(ix) = sec x	coth(ix) = -i cot x

PERIODICITY OF HYPERBOLIC FUNCTIONS

In the following k is any integer.
sinh (x + 2kπi) = sinh x     csch (x + 2kπi) = csch x
cosh (x + 2kπi) = cosh x     sech (x + 2kπi) = sech x
tanh (x + kπi) = tanh x     coth (x + kπi) = coth x

RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS

sin-1 (ix) = isinh-1x	sinh-1(ix) = i sin-1x
cos-1 x = ±i cosh-1 x	cosh-1x = ±i cos-1x
tan-1(ix) = i tanh-1x	tanh-1(ix) = i tan-1x
cot-1(ix) = -i coth-1x	coth-1 (ix) = -i cot-1x
sec-1 x = ±i sech-1x	sech-1 x = ±i sec-1x
csc-1(ix) = -i csch-1x	csch-1(ix) = -i csc-1x