Beautiful Trigonometry

Copyright c 20012004 Kevin Carmody

The unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3The unit hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The trigonometric hexagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Euler identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Exponential definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Inverse function log formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Product formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Ratio formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Inverse function reciprocal formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Pythagorean identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Pythagorean conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Inverse function Pythagorean conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Imaginary angle formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Inverse function imaginary angle formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Negative angle formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Inverse function negative angle formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Double angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Half angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Multiple-value identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Straight-angle translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Straight-angle reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Right-angle translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Right-angle reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Half-right-angle translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Inverse function multiple-value identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Inverse function straight-angle identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Inverse function right-angle identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Angle sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Function sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Function products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Inverse function sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Same-angle function sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19Linear combinations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20Multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Powers of two angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Powers of one angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Inverse function derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Inverse function power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Inner transformation function definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Gudermannian transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Gudermannian transformations of inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Gudermannian special transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Gudermannian special transformations of inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Cogudermannian and coangle transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Cogudermannian and coangle transformations of inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Properties of inner transformation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28Inner transformation conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Alternative inner transformation conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30DeMoivre identites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Point on the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Twice-applied formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2

The unit circle

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cos

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tan |

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y

The equation x2 + y2 = 1 is a circle of radius 1. This circle is called the unit circle.

For any point on the circle, twice the shaded area is the circular angle .

The indicated lengths are the circular trigonometric functions of .

3

The unit hyperbola

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cosh

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sinh ||||

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coth

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/2

x

y

The equation x2 y2 = 1 is a hyperbola bounded by a square and asymptotes on the corners of thesquare. This hyperbola is called the unit hyperbola.

For any point on the hyperbola, twice the shaded area is the hyperbolic angle .

The indicated lengths are the hyperbolic trigonometric functions of .

4

The trigonometric hexagon

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............. ............. ............. ............. ............. ............. ............. .............

1 cot

cossin

tan

sec csc

cg

cg

gd

gd

co

co

The trigonometric hexagon relates elementary properties of the six circular and hyperbolic trigono-metric functions. The points labelled sin, cos, tan, etc. are valid also for sinh, cosh, tanh, etc. The constant 1at the center is treated as a function.

Each function is the product of its two neighbors: e.g. sin = tan cos and sinh = tanh cosh .

Each function is the quotient of the two functions to the left and also of the two functions to the

right: e.g. sin =cos cot

=tan sec

.

The product of two opposite functions is 1, which means that opposite functions are reciprocals:

e.g. sin csc = 1, or sin =1

csc .

The three shaded triangles give the Pythagorean identities. For circular functions, the sum of thesquares of the top two functions is equal to the square of the bottom function; for hyperbolic functions,the sum of the squares of the left and the bottom functions is equal to the square of the right function. Forexample, sin2 + cos2 = 1; sinh2 + 1 = cosh2 .

The three dashed axes give the inner transformations. Inner transformations reflect functions fromone side of the axis to the other as follows:

The circular functions, when reflected across the co axis, become the circular functions on the otherside, e.g. sin co = cos .

The hyperbolic functions, when reflected across the cg axis, become the hyperbolic functions on theother side, e.g. sinh cg = csch .

Either circular or hyperbolic functions, when reflected across the gd axis, become the functionon the other side of the opposite type, i.e. circular becomes hyperbolic, and hyperbolic becomes circular.Examples: sinh gd = tanh ; sin gd1 = tan .

5

Euler identities

e = cosh + sinh = exp ei = cos + i sin = cis

e = cosh sinh = 1exp

ei = cos i sin = 1cis

Exponential definitions

cosh =e + e

2cos =

ei + ei

2

sinh =e e

2sin =

ei ei2i

tanh =e ee + e

tan = i ei eiei + ei

coth =e + e

e e cot = iei + ei

ei eisech =

2e + e

sec =2

ei + ei

csch =2

e e csc =2i

ei ei

Inverse function log formulae

cosh1 u = ln(u +u2 1

)cos1 u = i ln

(u i

1 u2

)sinh1 u = ln

(u +u2 + 1

)sin1 u = i ln

(1 u2 iu

)tanh1 u =

12

ln(

1 + u1 u

)tan1 u =

i

2ln(

1 iu1 + iu

)coth1 u =

12

ln(u + 1u 1

)cot1 u =

i

2ln(iu + 1iu 1

)sech1 u = ln

(1 +

1 u2u

)sec1 u = i ln

(1 iu2 1u

)

csch1 u = ln

(1 +

1 + u2

u

)csc1 u = i ln

(u2 1 iu

)exp1 u = lnu cis1 u = i lnu

6

Special values

0 + 0 pi2 pi4 pi6 pi3 pi10 pi5 3pi10 2pi5cosh + +1 + cos 1 0 1

2

3

212

2+2

2

32

1

2

sinh 0 + sin 0 1 12

12

3

21

2

32

2

2+2

tanh 1 0 +1 tan 0 1 13

3

7 47 + 4 3+45 3 + 4coth 1 +1 cot 0 1 3 1

3

3 + 4

3+4

5

7 + 4

7 4

sech 0 +1 0 sec 1

2 23

2 2

3 21 2

2+5 2

csch 0 0 csc 1

2 2 23

2 2

2+5 2

1 2

3 exp 1 +1 +

Signs

(, 0) (0,+) (0, pi2 ) (pi2 , pi) (pi, 3pi2 ) ( 3pi2 , 2pi)cosh + + cos + +sinh + sin + + tanh + tan + + coth + cot + + sech + + sec + +csch + csc + + exp + +

Ranges

R R+1 cosh < + 1 cos +1 < sinh < + 1 sin +11 < tanh < +1 < tan < +coth < 1 or +1 < coth < cot < +0 < sech +1 sec 1 or +1 sec < csch < +, but csch 6= 0 csc 1 or +1 csc 0 < exp | cis | = 1

7

Product formulae

cosh = sinh coth cos = sin cot sinh = cosh tanh sin = cos tan tanh = sinh sech tan = sin sec coth = cosh csch cot = cos csc sech = tanh csch sec = tan csc csch = coth sech csc = cot sec cosh sech = sinh csch cos sec = sin csc

= tanh coth = exp exp() = 1 = tan cot = cis cis() = 1

Ratio formulae

cosh =1

sech =

sinh tanh

=coth csch

cos =1

sec =

sin tan

=cot csc

sinh =1

csch =

cosh coth

=tanh sech

sin =1

csc =

cos cot

=tan sec

tanh =1

coth =

sinh cosh

=sech csch

tan =1

cot =

sin cos

=sec csc

coth =1

tanh =

cosh sinh

=csch sech

cot =1

tan =

cos sin

=csc sec

sech =1

cosh =

tanh sinh

=csch coth

sec =1

cos =

tan sin

=csc cot

csch =1

sinh =

coth cosh

=sech tanh

csc =1

sin =

cot cos

=sec tan

exp() = 1exp

cis() = 1cis

Inverse function reciprocal formulae

cosh1 u = sech11u

cos1 u = sec11u

sinh1 u = csch11u

sin1 u = csc11u

tanh1 u = coth11u

tan1 u = cot11u

coth1 u = tanh11u

cot1 u = tan11u

sech1 u = cosh11u

sec1 u = cos11u

csch1 u = sinh11u

csc1 u = sin11u

exp1 u = exp1 1u

cis1 u = cis1 1u

8

Pythagorean identities

1 = cosh2 sinh2 1 = cos2 + sin2 = tanh2 + sech2 = coth2 csch2 = sec2 tanh2 = csc2 cot2

cosh2 = sinh2 + 1 cos2 = 1 sin2 sinh2 = cosh2 1 sin2 = 1 cos2 tanh2 = 1 sech2 tan2 = sec2 1coth2 = csch2 + 1 cot2 = csc2 1sech2 = 1 tanh2 sec2 = 1 + tan2 csch2 = coth2 1 csc2 = cot2 + 1(coth tanh )2 = sech2 csch2 (cot + tan )2 = sec2 + csc2 coth tanh = sech csch cot + tan = sec csc sech2 csch2 = sech2 csch2 sec2 + csc2 = sec2 csc2

Pythagorean conversions

cosh =

1 + sinh2 =1

1 tanh2 cos =

1 sin2 = 1

1 + tan2

=coth

coth2 1=

csch2 1

csch =

cot 1 + cot2

=

csc2 1

csc

=exp2 + 1

2 exp =

cis2 + 12 cis

sinh =

cosh2 1 = tanh 1 tanh2

sin =

1 cos2 = tan 1 + tan2

=1

coth2 1=

1 sech2

sech =

11 + cot2

=

sec2 1

sec

=exp2 1

2 exp =

cis2 12i cis

tanh =

cosh2 1

cosh =

sinh 1 + sinh2

tan =

1 cos2

cos =

sin 1 sin2

=

1 sech2 = 1csch2 + 1

=

sec2 1 = 1csc2 1

=exp2 1exp2 + 1

= icis2 1

cis2 + 1

coth =cosh

cosh2 1=

1 + sinh2

sinh cot =

cos 1 cos2

=

1 sin2

sin

=1

1 sech2 =

csch2 + 1 =1

sec2 1=

csc2 1

=exp2 + 1exp2 1 = i

cis2 + 1

cis2 1

9

sech =1

1 + sinh2 =

1 tanh2 sec = 11 sin2

=

1 + tan2

=

coth2 1

coth =

csch csch2 1

=

1 + cot2

cot =

csc csc2 1

=2 exp

exp2 + 1=

2 cis

cis2 + 1

csch =1

cosh2 1=

1 tanh2

tanh csc =

11 cos2

=

1 + tan2

tan

=1

coth2 1=

sech 1 sech2

=

1 + cot2 =sec

sec2 1=

2 exp exp2 1 =

2i cis

cis2 1exp = cosh +

cosh2 1 cis = cos + i

1 cos2

=

sinh2 + 1 + sinh =

1 sin2 + i sin =

1 + tanh 1 tanh2

=coth + 1coth2 1

=1 + i tan

1 + tan2 =

cot + i1 + cot2

=1 +

1 sech2 sech

=

csch2 1 + 1

csch =

1 + i

sec2 1sec

=

csc2 1 + i

csc

Inverse function Pythagorean conversions

cosh1 u = sinh1u2 1 = tanh1

u2 1u

cos1 u = sin1

1 u2 = tan1

1 u2u

= coth1uu2 1

= csch11u2 1

= cot1u

1 u2= csc1

11 u2

= exp1(u +u2 1

)= cis1

(u + i

1 u2

)sinh1 u = cosh1

1 + u2 = tanh1

u1 + u2

sin1 u = cos1

1 u2 = tan1 u1 u2

= coth1

1 + u2

u= sech1

11 + u2

= cot1

1 u2u

= sec11

1 u2= exp1

(u +

1 + u2)

= cis1(

1 u2 + iu)

tanh1 u = cosh11

1 u2= sinh1

u1 u2

tan1 u = cos11

1 + u2= sin1

u1 + u2

= sech1

1 u2 = csch1

1 u2u

= sec1

1 + u2 = csc1

1 + u2

u

= exp1

1 + u1 u = cis

1

1 + iu1 iu

coth1 u = cosh1uu2 1

= sinh11u2 1

cot1 u = cos1u

1 + u2= sin1

11 + u2

= sech1u2 1u

= csch1u2 1 = sec1

1 + u2

u= csc1

1 + u2

= exp1u 1u + 1

= cis1iu 1iu + 1

10

sech1 u = sinh1

1 u2u

= tanh1

1 u2 sec1 u = sin1u2 1u

= tan1u2 1

= coth11

1 u2= csch1

u1 u2

= cot11u2 1

= csc1uu2 1

= exp11 +u2 1u

= cis11 + iu2 1u

csch1 u = cosh1u2 + 1u

= tanh11u2 + 1

csc1 u = cos1u2 1u

= tan11u2 1

= coth1u2 + 1 = sech1

uu2 + 1

= cot1u2 1 = sec1 u

u2 1= exp1

1 +

1 + u2

u= cis1

u2 1 + iu

exp1 u = cosh1u2 + 1

2u= sinh1

u2 12u

cis1 u = cos1u2 + 1

2u= sin1

u2 12iu

= tanh1u2 1u2 + 1

= coth1u2 + 1u2 1 = tan

1 u2 1

i(u2 + 1)= cot1

i(u2 + 1)u2 1

= sech12u

u2 + 1= csch1

2uu2 1 = sec

1 2uu2 + 1

= csc12iuu2 1

Imaginary angle formulae

cosh i = cos cos i = cosh sinh i = i sin sin i = i sinh tanh i = i tan tan i = i tanh coth i = i cot cot i = i coth sech i = sec sec i = sech csch i = i csc csc i = i csch exp i = cis cis i =

1exp

exp = (cis )i cis =(exp

)i

Inverse function imaginary angle formulae

i cosh1 u = cos1 u i cos1 u = cosh1 ui sinh1 u = sin1 iu i sin1 u = sinh1 iui tanh1 u = tan1 iu i tan1 u = tanh1 iui coth1 u = cot1 iu i cot1 u = coth1 iui sech1 u = sec1 u i sec1 u = sech1 ui csch1 u = csc1 iu i csc1 u = csch1 iu

11

Negative angle formulae

cosh() = cosh cos() = cos sinh() = sinh sin() = sin tanh() = tanh tan() = tan coth() = coth cot() = cot sech() = sech sec() = sec csch() = csch csc() = csc exp() = 1

exp cis() = 1

cis

Inverse function negative angle formulae

cosh1(u) = pii cosh1 u cos1(u) = pi cos1 usinh1(u) = sinh1 u sin1(u) = sin1 utanh1(u) = tanh1 u tan1(u) = tan1 ucoth1(u) = coth1 u cot1(u) = cot1 usech1(u) = pii sech1 u sec1(u) = pi sec1 ucsch1(u) = csch1 u csc1(u) = csc1 uexp1(u) = pii + exp1 u cis1(u) = pi + cis1 u

Double angles

cosh 2 = cosh2 + sinh2 cos 2 = cos2 sin2 = 2 cosh2 1 = 1 + 2 sinh2 = 2 cos2 1 = 1 2 sin2

sinh 2 = 2 cosh sinh sin 2 = 2 cos sin = i(

1 (cosh + i sinh )2)

= (cos + sin )2 1

tanh 2 =2 tanh

1 + tanh2 tan 2 =

2 tan 1 tan2

coth 2 =coth2 + 1

2 coth cot 2 =

cot2 12 cot

sech 2 =csch2 sech2

csch2 + sech2 sec 2 =

csc2 sec2 csc2 sec2

csch 2 =csch sech

2csc 2 =

csc sec 2

exp 2 = exp2 cis 2 = cis2

12

Half angles

cosh

2=

cosh + 12

cos

2=

1 + cos 2

sinh

2=

cosh 12

sin

2=

1 cos 2

tanh

2=

sinh cosh + 1

=cosh 1

sinh = coth csch tan

2=

sin 1 + cos

=1 cos

sin = csc cot

coth

2=

sinh cosh 1 =

cosh + 1sinh

= coth + csch cot

2=

sin 1 cos =

1 + cos sin

= csc + cot

sech

2=

2 sech

1 + sech sec

2=

2 sec

sec + 1

csch

2=

2 sech

1 sech csc

2=

2 sec

sec 1exp

2=

exp cis

2=

cis

Multiple-value identities

cosh( + 2pii) = cosh cos( + 2pi) = cos sinh( + 2pii) = sinh sin( + 2pi) = sin tanh( + 2pii) = tanh tan( + 2pi) = tan coth( + 2pii) = coth cot( + 2pi) = cot sech( + 2pii) = sech sec( + 2pi) = sec csch( + 2pii) = csch csc( + 2pi) = csc exp( + 2pii) = exp cis( + 2pi) = cis

Straight-angle translations

cosh( + pii) = cosh( pii) = cosh cos( + pi) = cos( pi) = cos sinh( + pii) = sinh( pii) = sinh sin( + pi) = sin( pi) = sin tanh( + pii) = tanh( pii) = tanh tan( + pi) = tan( pi) = tan coth( + pii) = coth( pii) = coth cot( + pi) = cot( pi) = cot sech( + pii) = sech( pii) = sech sec( + pi) = sec( pi) = sec csch( + pii) = csch( pii) = csch csc( + pi) = csc( pi) = csc exp( + pii) = exp( pii) = exp cis( + pi) = cis( pi) = cis

13

Straight-angle reflections

cosh(pii ) = cosh cos(pi ) = cos sinh(pii ) = sinh sin(pi ) = sin tanh(pii ) = tanh tan(pi ) = tan coth(pii ) = coth cot(pi ) = cot sech(pii ) = sech sec(pi ) = sec csch(pii ) = csch csc(pi ) = csc exp(pii ) = 1

exp cis(pi ) = 1

cis

Right-angle translations

cosh( + pii2 ) = i sinh cos( +pi2 ) = sin

sinh( + pii2 ) = i cosh sin( +pi2 ) = cos

tanh( + pii2 ) = coth tan( +pi2 ) = cot

coth( + pii2 ) = tanh cot( +pi2 ) = tan

sech( + pii2 ) = i csch sec( + pi2 ) = csc csch( + pii2 ) = i sech csc( + pi2 ) = sec exp( + pii2 ) = i exp cis( +

pi2 ) = i cis

cosh( pii2 ) = i sinh cos( pi2 ) = sin sinh( pii2 ) = i cosh sin( pi2 ) = cos tanh( pii2 ) = coth tan( pi2 ) = cot coth( pii2 ) = tanh cot( pi2 ) = tan sech( pii2 ) = i csch sec( pi2 ) = csc csch( pii2 ) = i sech csc( pi2 ) = sec exp( pii2 ) = i exp cis( pi2 ) = i cis

Right-angle reflections

cosh(pii2 ) = i sinh cos(pi2 ) = sin sinh(pii2 ) = i cosh sin(pi2 ) = cos tanh(pii2 ) = coth tan(pi2 ) = cot coth(pii2 ) = tanh cot(pi2 ) = tan sech(pii2 ) = i csch sec(pi2 ) = csc csch(pii2 ) = i sech csc(pi2 ) = sec exp(pii2 ) =

i

exp cis(pi2 ) =

i

cis

14

Half-right-angle translations

cosh( +

pii

4

)= i sinh

(pii

4 )

cos( +

pi

4

)= sin

(pi4 )

=cosh + i sinh

2=

cos sin 2

sinh( +

pii

4

)= i cosh

(pii

4 )

sin( +

pi

4

)= cos

(pi4 )

=i cosh + sinh

2=

cos + sin 2

tanh( +

pii

4

)= coth

(pii

4 )

tan( +

pi

4

)= cot

(pi4 )

=i + tanh

1 + i tanh =

1 + tan 1 tan

coth( +

pii

4

)= tanh

(pii

4 )

cot( +

pi

4

)= tan

(pi4 )

=coth + ii coth + 1

=cot 1cot + 1

sech( +

pii

4

)= i csch

(pii

4 )

sec( +

pi

4

)= csc

(pi4 )

=2

2 csch 2csch + i sech

=2

2 csc 2csc sec

csch( +

pii

4

)= i sech

(pii

4 )

csc( +

pi

4

)= sec

(pi4 )

=2

2 csch 2i csch + sech

=2

2 csc 2csc + sec

exp( +

pii

4

)=

1

exp(pii

4 ) cis( + pi

4

)=

i

cis(pi

4 )

=(1 + i) exp

2=

(1 + i) cis 2

Inverse function multiple-value identities

cosh1 u = 2pii + i cosh1 u cos1 u = 2pi + cos1 usinh1 u = 2pii + i sinh1 u sin1 u = 2pi + sin1 utanh1 u = pii + i tanh1 u tan1 u = pi + tan1 ucoth1 u = pii + i coth1 u cot1 u = pi + cot1 usech1 u = 2pii + i sech1 u sec1 u = 2pi + sec1 ucsch1 u = 2pii + i csch1 u csc1 u = 2pi + csc1 u

15

Inverse function straight-angle identities

cosh1 u = pii i cosh1 u cos1 u = pi cos1 usinh1 u = pii i sinh1 u sin1 u = pi sin1 utanh1 u = pii + i tanh1 u tan1 u = pi + tan1 ucoth1 u = pii + i coth1 u cot1 u = pi + cot1 usech1 u = pii i sech1 u sec1 u = pi sec1 ucsch1 u = pii i csch1 u csc1 u = pi csc1 u

Inverse function right-angle identities

cosh1 u =pii

2 i sinh1 u cos1 u = pi

2 sin1 u

sinh1 u =pii

2+ i cosh1 u sin1 u =

pi

2+ cos1 u

tanh1 u =pii

2+ coth1 u tan1 u =

pi

2 cot1 u

coth1 u =pii

2+ tanh1 u cot1 u =

pi

2 tan1 u

sech1 u =pii

2 i csch1 u sec1 u = pi

2 csc1 u

csch1 u =pii

2 i sech1 u csc1 u = pi

2+ sec1 u

Angle sums and differences

cosh(a + b) = cosha cosh b + sinha sinh b cos(a + b) = cosa cos b sina sin bcosh(a b) = cosha cosh b sinha sinh b cos(a b) = cosa cos b + sina sin bsinh(a + b) = sinha cosh b + cosha sinh b sin(a + b) = sina cos b + cosa sin bsinh(a b) = sinha cosh b cosha sinh b sin(a b) = sina cos b cosa sin btanh(a + b) =

tanha + tanh b1 + tanha tanh b

tan(a + b) =tana + tan b

1 tana tan btanh(a b) = tanha tanh b

1 tanha tanh b tan(a b) =tana tan b

1 + tana tan b

coth(a + b) =cotha coth b + 1cotha + coth b

cot(a + b) =cota cot b 1cota + cot b

coth(a b) = cotha coth b 1coth b cotha cot(a b) =

cota cot b + 1cot b cota

sech(a + b) =cscha csch b secha sech b

cscha csch b + secha sech bsec(a + b) =

csca csc b seca sec bcsca csc b seca sec b

sech(a b) = cscha csch b secha sech bcscha csch b secha sech b sec(a b) =

csca csc b seca sec bcsca csc b + seca sec b

csch(a + b) =secha csch b cscha sech b

secha csch b + cscha sech bcsc(a + b) =

seca csc b csca sec bseca csc b + csca sec b

csch(a b) = secha csch b cscha sech bsecha csch b cscha sech b csc(a b) =

seca csc b csca sec bseca csc b csca sec b

exp(a + b) = expa exp b cis(a + b) = cisa cis b

exp(a b) = expaexp b

cis(a b) = cisacis b

16

Function sums and differences

cosha + cosh b = 2 cosha + b

2cosh

a b2

cosa + cos b = 2 cosa + b

2cos

a b2

cosha cosh b = 2 sinh a + b2

sinha b

2cosa cos b = 2 sin b + a

2sin

b a2

sinha + sinh b = 2 sinha + b

2cosh

a b2

sina + sin b = 2 sina + b

2cos

a b2

sinha sinh b = 2 cosh a + b2

sinha b

2sina sin b = 2 cos a + b

2sin

a b2

tanha + tanh b = sinh(a + b) secha sech b tana + tan b = sin(a + b) seca sec btanha tanh b = sinh(a b) secha sech b tana tan b = sin(a b) seca sec bcotha + coth b = sinh(a + b) cscha csch b cota + cot b = sin(a + b) csca csc bcotha coth b = sinh(b a) cscha csch b cota cot b = sin(b a) csca csc bsecha + sech b = 2 cosh

a + b2

cosha b

2secha sech b seca + sec b = 2 cos

a + b2

cosa b

2seca sec b

secha sech b = 2 sinh b + a2

sinhb a

2secha sech b seca sec b = 2 sin a + b

2sin

a b2

seca sec b

cscha + csch b = 2 sinha + b

2cosh

a b2

cscha csch b csca + csc b = 2 sina + b

2cos

a b2

csca csc b

cscha csch b = 2 cosh b + a2

sinhb a

2cscha csch b csca csc b = 2 cos b + a

2sin

b a2

csca csc b

expa + exp b = 2 cosha b

2exp

a + b2

cisa + cis b = 2 cosa b

2cis

a + b2

expa exp b = 2 sinh a b2

expa + b

2cisa cis b = 2i sin b a

2cis

a + b2

cosha + i sinh b cosa + sin b

= 2 sinh(a b

2 pii

4

)sinh

(a + b

2+pii

4

)= 2 sin

(pi

4 a b

2

)sin(pi

4+a + b

2

)cosha i sinh b cosa sin b= 2 sinh

(a + b

2 pii

4

)sinh

(a b

2+pii

4

)= 2 sin

(pi

4 a + b

2

)sin(pi

4+a b

2

)cotha + tanh b = cosh(a + b) cscha sech b cota + tan b = cos(a b) csca sec bcotha tanh b = cosh(a b) cscha sech b cota tan b = cos(a + b) csca sec bcscha + i sech b = 2 cscha sech b csca + sec b = 2 csca sec b

sinh(b a

2 pii

4

)sinh

(b + a

2+pii

4

)sin(pi

4 b a

2

)sin(pi

4+b + a

2

)cscha i sech b = 2 cscha sech b csca + sec b = 2 csca sec b

sinh(b + a

2 pii

4

)sinh

(b a

2+pii

4

)sin(pi

4 b + a

2

)sin(pi

4+b a

2

)sinha + sinh bcosha + cosh b

=cosha cosh bsinha sinh b = tanh

a + b2

sina + sin bcosa + cos b

=cos b cosasina sin b = tan

a + b2

sinha sinh bcosha + cosh b

=cosha cosh bsinha + sinh b

= tanha b

2sina sin bcosa + cos b

=cos b cosasina + sin b

= tana b

2tanha + tanh btanha tanh b =

cotha + coth bcoth b cotha

tana + tan btana tan b =

cot b + cotacot b cota

= sinh(a + b) csch(a b) = sin(a + b) csc(a b)

17

secha sech bsecha + sech b

= tanha + b

2tanh

b a2

seca sec bseca + sec b

= tana + b

2tan

a b2

cscha + csch bcscha csch b = tanh

a + b2

cothb a

2csca + csc bcsca csc b = tan

a + b2

cotb a

2expa exp bexpa + exp b

= 2 tanha b

2cisa cis bcisa + cis b

= 2i tanb a

2

Function products

cosha cosh b =cosh(a + b) + cosh(a b)

2cosa cos b =

cos(a b) + cos(a + b)2

sinha cosh b =sinh(a + b) + sinh(a b)

2sina cos b =

sin(a b) + sin(a + b)2

sinha sinh b =cosh(a + b) cosh(a b)

2sina sin b =

cos(a b) cos(a + b)2

tanha tanh b = coth(a + b)(tanha + tanh b) 1 tana tan b = 1 cot(a + b)(tana + tan b)tanha coth b = tanh(a + b)(cotha + tanh b) 1 tana cot b = tan(a + b)(cot b tana) 1cotha coth b = coth(a + b)(cotha + coth b) 1 cota cot b = cot(a + b)(cota + cot b) + 1secha sech b = sech(a + b)(1 + tanha tanh b) seca sec b = sec(a + b)(1 tana tan b)

= csch(a + b)(tanha + tanh b) = csch(a + b)(tana + tan b)secha csch b = sech(a + b)(coth b + tanha) seca csc b = sec(a + b)(cot b tana)

= csch(a + b)(tanha coth b + 1) = csch(a + b)(tana cot b + 1)cscha csch b = sech(a + b)(cotha coth b 1) csca csc b = sec(a + b)(cota cot b 1)

= csch(a + b)(cotha + coth b) = csch(a + b)(cota + cot b)expa exp b = exp(a + b) cisa cis b = cis(a + b)

Inverse function sums and differences

cosh1 u + cosh1 v cos1 u + cos1 v= cosh1

(uv +

u2 1

v2 1

)= cos1

(uv

1 u2

1 v2

)= sinh1

(uv2 1 + v

u2 1

)= sin1

(u

1 v2 + v

1 u2)

cosh1 u cosh1 v cos1 u cos1 v= cosh1

(uv

u2 1

v2 1

)= cos1

(uv +

1 u2

1 v2

)= sinh1

(vu2 1 u

v2 1

)= sin1

(v

1 u2 u

1 v2)

sinh1 u + sinh1 v sin1 u + sin1 v= sinh1

(uv2 + 1 + v

u2 + 1

)= sin1

(u

1 v2 + v

1 u2)

= cosh1(

u2 + 1v2 + 1 + uv

)= cos1

(1 u2

1 v2 uv

)sinh1 u sinh1 v sin1 u sin1 v

= sinh1(uv2 + 1 v

u2 + 1

)= sin1

(u

1 v2 v

1 u2)

= cosh1(

u2 + 1v2 + 1 uv

)= cos1

(1 u2

1 v2 + uv

)tanh1 u + tanh1 v = tanh1

u + v1 + uv

tan1 u + tan1 v = tan1u + v

1 uvtanh1 u tanh1 v = tanh1 u v

1 uv tan1 u tan1 v = tan1 u v

1 + uv

18

coth1 u + coth1 v = coth11 + uvu + v

cot1 u + cot1 v = cot1uv 1u + v

coth1 u coth1 v = coth1 1 uvu v cot

1 u cot1 v = cot1 vu 1v u

sech1 u + sech1 v sec1 u + sec1 v

= sech1uv

1 u2

1 v2uv +

1 u2

1 v2

= sec1uvu2 1

v2 1

u2 1v2 1 uv

= csch1uv

1 u2

1 v2u

1 v2 + v

1 u2= csc1

uvu2 1

v2 1

uv2 1 + v

u2 1

sech1 u sech1 v sec1 u sec1 v= sech1

uv

1 u2

1 v21 u2

1 v2 uv

= sec1uvu2 1

v2 1

uv +u2 1

v2 1

= csch1uv

1 u2

1 v2u

1 v2 v

1 u2= csc1

uvu2 1

v2 1

uv2 1 v

u2 1

csch1 u + csch1 v csc1 u + csc1 v

= csch1uvu2 + 1

v2 + 1

uv2 + 1 + v

u2 + 1

= csc1uvu2 1

v2 1

uv2 1 + v

u2 1

= sech1uvu2 + 1

v2 + 1

uv +u2 + 1

v2 + 1

= sec1uvu2 1

v2 1

uv u2 1

v2 1

csch1 u csch1 v csc1 u csc1 v= csch1

uvu2 + 1

v2 + 1

vu2 + 1 u

v2 + 1

= csc1uvu2 1

v2 1

vu2 1 u

v2 1

= sech1uvu2 + 1

v2 + 1

uv u2 + 1

v2 + 1

= sec1uvu2 1

v2 1

uv +u2 1

v2 1

exp1 u + exp1 v = exp1 uv cis1 u + cis1 v = cis1 uv

exp1 u exp1 v = exp1 uv

cis1 u cis1 v = cis1 uv

Same-angle function sums and differences

cosh + i sinh =

2i

sinh( +

pii

4

)cos + sin =

2 sin

( +

pi

4

)=

2 cosh(pii

4 )

=

2 cos(pi

4 )

cosh i sinh =

2 cosh( +

pii

4

)cos sin =

2 cos

( +

pi

4

)=

2i

sinh(pii

4 )

=

2 sin(pi

4 )

coth + tanh = 2 coth 2 cot + tan = 2 csc 2coth tanh = 2 csch 2 cot tan = 2 cot 2

19

csch + i sech =

2 csch

( +

pii

4

)i csch 2

csc + sec =

2 csc

( +

pi

4

)csc 2

=

2 sech

(pii

4 )

csch 2=

2 sec

(pi4 )

csc 2

csch i sech =

2 sech

( +

pii

4

)i csch 2

csc sec =

2 sec( +

pi

4

)csc 2

=

2 csch

(pii

4 )

csch 2=

2 csc

(pi4 )

csc 2

Linear combinations of functions

A 6= B A 6= B A sinh + B cosh A sin + B cos

=A2 B2 sinh

( + tanh1

B

A

)=A2 + B2 sin

( + tan1

B

A

)=B2 A2 cosh

( + tanh1

A

B

)=A2 + B2 cos

( tan1 A

B

)A tanh + B coth A tan + B cot

=A2 sech2 B2 csch2 =

A2 sec2 + B2 csc2

sinh( + tanh1

(B

Acoth

))sin( + tan1

(B

Acot

))=B2 csch2 A2 sech2 =

A2 sec2 + B2 csc2

cosh( + tanh1

(A

Btanh

))cos( tan1

(A

Btan

))A sech + B csch A sec + B csc

=A2 B2 csch 2 sinh

( + tanh1

B

A

)=A2 + B2 csc 2 sin

( + tan1

B

A

)=B2 A2 csch 2 cosh

( + tanh1

A

B

)=A2 + B2 csc 2 cos

( tan1 A

B

)A 6= B and A,B 6= 0 A 6= B and A,B 6= 0A exp +

B

exp A cis +

B

cis

= 2AB sinh

( + tanh1

A BA + B

)= 2AB sin

( + tan1

A BA + B

)= 2iAB cosh

( + tanh1

A + BA B

)= 2AB cos

( tan1 A + B

A B)

Multiple angles

coshn =

n2

k=0

(n

2k

)coshn2k sinh2k cosn =

n2

k=0

(1)k(n

2k

)cosn2k sin2k

sinhn =

n12

k=0

(n

2k + 1

)coshn2k1 sinh2k+1 sinn =

n12

k=0

(1)k(

n

2k + 1

)cosn2k1 sin2k+1

20

Powers of two angles

sinh2 a sinh2 b = cosh2 a cosh2 b sin2 a sin2 b = cos2 b cos2 a= sinh(a + b) sinh(a b) = sin(a + b) sin(a b)

cosh2 a + sinh2 b = cosh2 b + sinh2 a cos2 a sin2 b = cos2 b sin2 a= cosh(a + b) cosh(a b) = cos(a + b) cos(a b)

tanh2 a tanh2 b tan2 a tan2 b= tanh(a + b) tanh(a b)(1 tanh2 a tanh2 b) = tan(a + b) tan(a b)(1 tan2 a tan2 b)

coth2 a coth2 b cot2 a cot2 b= tanh(a + b) tanh(a b)(1 coth2 a coth2 b) = tan(a + b) tan(a b)(1 cot2 a cot2 b)

sech2 a sech2 b sec2 a sec2 b= sinh(a + b) sinh(a b) sech2 a sech2 b = sin(a + b) sin(a b) sec2 a sec2 b

csch2 a csch2 b csc2 a csc2 b= sinh(b + a) sinh(b a) csch2 a csch2 b = sin(b + a) sin(b a) csc2 a csc2 b

Powers of one angle

cosh2n cos2n

=14n

(2nn

)=

14n

(2nn

)

+1

22n1

nk=1

(2nn + k

)cosh [2k] +

122n1

nk=1

(2nn + k

)cos [2k]

sinh2n sin2n

=14n

(1)n(

2nn

)=

14n

(2nn

)

+1

22n1

nk=1

(1)n+k(

2nn + k

)cosh [2k] +

122n1

nk=1

(1)k(

2nn + k

)cos [2k]

cosh2n+1 cos2n+1

=1

4n1

nk=0

(2n + 1n + k + 1

)cosh [(2k + 1)] =

14n1

nk=0

(2n + 1n + k + 1

)cos [(2k + 1)]

sinh2n+1 sin2n+1

=1

4n1

nk=0

(1)k+1(

2n + 1n + k + 1

)sinh [(2k + 1)] =

14n1

nk=0

(1)k(

2n + 1n + k + 1

)sin [(2k + 1)]

coshn sinhn cosn sinn

=nk=1

1 exp 4kpii

n =

nk=1

1 + cis

4kpin

sinh( tanh1 exp 4kpii

n

)sin( tan1 cis 4kpi

n

)

21

coshn + sinhn cosn + sinn

=nk=1

1 exp 2(2k + 1)pii

n =

nk=1

1 + cis

2(2k + 1)pin

sinh( tanh1 exp 2(2k + 1)pii

n

)sin( tanh1 cis 2(2k + 1)pi

n

)cosh2n sinh2n cos2n sin2n

=1

22n1(1 (1)n)

(2nn

)+

14n1 = 1

4n1

n2

k=1

(2n

n + 2k + (1)n)

cosh [(2k + (1)n) 2]n+1

2k=1

(2n

n + 2k 1

)cos [(2k 1) 2]

cosh2n + sinh2n cos2n + sin2n

=1

22n1(1 + (1)n)

(2nn

)+

14n1 = 1

22n1

(2nn

)+

14n1

n+12

k=1

(2n

n + 2k (1)n)

cosh [(2k (1)n) 2]n2

k=1

(2n

n + 2k

)cos [4k]

cosh2n+1 i sinh2n+1 cos2n+1 sin2n+1 =

122n3

= 122n3

nk=0

(2n + 1n + k + 1

)cosh

[pii

4+ (1)k(2k + 1)

] nk=0

(2n + 1n + k + 1

)cos[pi

4+ (1)k(2k + 1)

]cosh2n+1 + i sinh2n+1 cos2n+1 + sin2n+1

= i1

22n3 = 1

22n3

nk=0

(2n + 1n + k + 1

)sinh

[pii

4+ (1)k(2k + 1)

] nk=0

(2n + 1n + k + 1

)sin[pi

4+ (1)k(2k + 1)

]cosh2n+1 + sinh2n+1 cos2n+1 + i sin2n+1

=1

4n1 = 1

4n1

nk=0

(2n + 1n + k + 1

)exp[(1)k+1(2k + 1)

] nk=0

(2n + 1n + k + 1

)cis[(1)k(2k + 1)

]

Differential equations

f () = f() f() = A cosh + B sinh f () = f() f() = A cos + B sin

Derivatives

(cosh ) = sinh (cos ) = sin (sinh ) = cosh (sin ) = cos (tanh ) = sech2 (tan ) = sec2 (coth ) = csch2 (cot ) = csc2 (sech ) = tanh sech (sec ) = tan sec (csch ) = coth csch (csc ) = cot csc (exp ) = exp (cis ) = i cis

22

Inverse function derivatives

(cosh1 u

)=

1u2 1

(cos1 u

) = 11 u2(

sinh1 u)

=1u2 + 1

(sin1 u

)=

11 u2(

tanh1 u)

=1

1 u2(tan1 u

) = 11 + u2(

coth1 u)

=1

1 u2(cot1 u

) = 11 + u2(

sech1 u)

=1

u

1 u2(sec1 u

) = 1uu2 1(

csch1 u)

=1

u

1 + u2(csc1 u

) = 1uu2 1(

exp1 u) = 1

u

(cis1 u

)=iu

Power series

exp =k=0

k

k!cis =

k=0

(i)k

k!

cosh =k=0

2k

(2k)!cos =

k=0

(1)k2k(2k)!

sinh =k=0

2k+1

(2k + 1)!sin =

k=0

(1)k2k+1(2k + 1)!

Bk =2(2k)!

(4k 1)pi2kj=0

1(2j + 1)2k

Ek =4k+1(2k)!pi2k+1

j=0

(1)j+1(2j 1)2k+1

tanh =k=1

(1)k4k(4k 1)Bk(2k)!

2k1, tan =k=1

4k(4k 1)Bk(2k)!

2k1,

|| < pi2 || < pi2coth =

1k=1

(1)k+14kBk(2k)!

2k1, cot =1k=1

4kBk(2k)!

2k1,

0 < || < pi 0 < || < pisech =

k=0

(1)kEk(2k)!

2k , sec =k=0

Ek(2k)!

2k ,

|| < pi2 || < pi2csch =

1+k=1

(1)k2(22k1 1)Bk(2k)!

2k1, csc =1+k=1

2(22k 1)Bk(2k)!

2k1,

|| < pi || < pi

23

Inverse function power series

sinh1 u =k=0

(1)k(2k)!4k(k!)2(2k + 1)

u2k+1, sin1 u =k=0

(2k)!4k(k!)2(2k + 1)

u2k+1,

|u| < 1 |u| < 1cosh1 u = ln 2u

k=1

(2k)!4k(k!)22k

u2k , cos1 u =pi

2k=0

(2k)!4k(k!)2(2k + 1)

u2k+1,

|u| > 1 |u| < 1tanh1 u =

k=0

u2k+1

2k + 1, |u| < 1 tan1 u =

k=0

(1)ku2k+12k + 1

, |u| 1

coth1 u =k=0

u2k1

2k + 1, |u| > 1 cot1 u =

k=0

(1)ku2k12k + 1

, |u| 1

sech1 u = ln2uk=1

(2k)!4k(k!)22k

u2k , sec1 u =pi

2k=0

(2k)!4k(k!)2(2k + 1)

u2k1,

|u| < 1 |u| > 1csch1 u =

k=0

(1)k(2k)!4k(k!)2(2k + 1)

u2k1, csc1 u =k=0

(2k)!4k(k!)2(2k + 1)

u2k1,

|u| > 1 |u| > 1

Infinite products

cosh =k=1

1 +(

pi(k 12 ))2

cos =k=1

1 (

pi(k 12 ))2

sinh = k=1

1 +(

pik

)2sin =

k=1

1 (

pik

)2

tanh = k=1

(1 1

2k

)2 k2 +( pi)2

(k 12 )2 +(

pi

)2 tan = k=1

(1 1

2k

)2 k2 ( pi)2

(k 12 )2 (

pi

)2

coth =1

k=1

(2k

2k 1)2 (k 12 )2 +( pi

)2k2 +

(

pi

)2 cot = 1k=1

(2k

2k 1)2 (k 12 )2 ( pi

)2k2

(

pi

)2sech =

k=1

(k 12 )2

(k 12 )2 +(

pi

)2 sec = k=1

(k 12 )2

(k 12 )2 (

pi

)2csch =

k=1

k2

k2 +(

pi

)2 csc = k=1

k2

k2 (

pi

)22e=

k=1

exp( 1

2k(2k + 1)

)2pi

=k=2

cos(pi

2k

)

24

Continued fractions

exp =1

1 1 +

2 3 +

2 5 +

2 . . .

cis =1

i +

i 2i +

3i 2i +

5i 2i +

. . .

tanh =

1 +2

3 +2

5 +2

7 +2

9 +2

11 +2

13 +2

. . .

tan =

1 2

3 2

5 2

7 2

9 2

11 2

13 2

. . .

coth =1+

3 +2

5 +2

7 +2

9 +2

11 +2

13 +2

. . .

cot =1

3 2

5 2

7 2

9 2

11 2

13 2

. . .

exp1 u =u

1 +u

2 +u

3 +4u

4 +4u

5 +9u

6 +9u

7 +16u. . .

cis1 u =u

i u2i u

3i 4u4i 4u

5i 9u6i 9u

7i 16u. . .

25

tanh1 u =u

1 u2

3 4u2

5 9u2

7 16u2

9 25u2

11 36u2

. . .

tan1 u =u

1 +u2

3 +4u2

5 +9u2

7 +16u2

9 +25u2

11 +36u2

. . .

coth1 u =1

u 13u 4

5u 97u 16

9u 2511u 36

. . .

cot1 u =1

u +1

3u +4

5u +9

7u +16

9u +25

11u +36. . .

Inner transformation function definitions

gd = 2 tan1 e pi2

=pi

2 2 cot1 e gd1 = ln tan

(

2+pi

4

)= ln cot

(pi

4

2

)= ln(sec + tan )

cg = 2 coth1 e = 2 tanh1 e co = pi2 = pi

2 + 2npi , n N

= ln(csch + coth )

gs =gd 2

2gs1 =

gd1 22

Gudermannian transformations

cosh gd1 = sec cos gd = sech sinh gd1 = tan sin gd = tanh tanh gd1 = sin tan gd = sinh coth gd1 = csc cot gd = csch sech gd1 = cos sec gd = cosh csch gd1 = cot csc gd = coth

26

Gudermannian transformations of inverses

gd cosh1 u = sec1 u gd1 cos1 u = sech1 ugd sinh1 u = tan1 u gd1 sin1 u = tanh1 ugd tanh1 u = sin1 u gd1 tan1 u = sinh1 ugd coth1 u = csc1 u gd1 cot1 u = csch1 ugd sech1 u = cos1 u gd1 sec1 u = cosh1 ugd csch1 u = cot1 u gd1 csc1 u = coth1 u

Gudermannian special transformations

tanh gs1 = tan tan gs = tanh coth gs1 = cot cot gs = coth

Gudermannian special transformations of inverses

gs tanh1 u = tan1 u gs1 tan1 u = tanh1 ugs coth1 u = cot1 u gs1 cot1 u = coth1 u

Cogudermannian and coangle transformations

cosh cg = coth cos co = sin sinh cg = csch sin co = cos tanh cg = sech tan co = cot coth cg = cosh cot co = tan sech cg = tanh sec co = csc csch cg = sinh csc co = sec

Cogudermannian and coangle transformations of inverses

cg cosh1 u = coth1 u co cos1 u = sin1 ucg sinh1 u = csch1 u co sin1 u = cos1 ucg tanh1 u = sech1 u co tan1 u = cot1 ucg coth1 u = cosh1 u co cot1 u = tan1 ucg sech1 u = tanh1 u co sec1 u = csc1 ucg csch1 u = sinh1 u co csc1 u = sec1 u

27

Properties of inner transformation functions

gd() = gd gd1() = gd1 cg() = cg pii2 co() = pi co gs() = gs gs1() = gs1 (gd ) = sech (gd1 ) = sec (cg ) = csch (co ) = (gs ) = sech 2 (gs1 ) = sec 2gd(i) = i gd1 gd1(i) = i gd cg1 = cg co1 = co gs(i) = i gs1 gs1(i) = i gs

gd(a + b) = tan1sinhacosh b

+ tan1sinh bcosha

gd1(a + b) = tanh1sinacos b

+ tanh1sin bcosa

gd(a b) = tan1 sinhacosh b

tan1 sinh bcosha

gd1(a b) = tanh1 sinacos b

tanh1 sin bcosa

cg(a + b) = co(a + b) =

tanh1cosh b sinhacosh b + sinha

+ tanh1cosha sinh bcosha + sinh b

co 2a2

+co 2b

2cg(a b) = co(a b) =

tanh1cosh b sinhacosh b + sinha

+ tanh1cosha + sinh bcosha sinh b coa + b

gs(a + b) = 12 tan1 2 cosha sinha

cosh2 b + sinh2 bgs1(a + b) = 12 tanh

1 2 cosa sina

cos2 b + sin2 b

+ 12 tan1 2 cosh b sinh b

cosh2 a + sinh2 a+ 12 tanh

1 2 cos b sin b

cos2 a + sin2 a

gs(a b) = 12 tan12 cosha sinha

cosh2 b + sinh2 bgs1(a b) = 12 tanh1

2 cosa sina

cos2 b + sin2 b

12 tan12 cosh b sinh b

cosh2 a + sinh2 a 12 tanh1

2 cos b sin b

cos2 a + sin2 agd 2 = 2 tan1 tanh gd1 2 = 2 tanh1 tan cg 2 = 2 coth1 exp 2 co 2 = 2 co pi2gs 2 = tan1

2 tanh

1 + tanh2 gs1 2 = tanh1

2 tan 1 tan2

gd

2= csc1(csch + coth ) gd1

2= csch1(csc + cot )

cg

2= cosh1(csch + coth ) co

2= tan1(csc + cot )

gs

2=

tan1 sin 2

gs1

2=

tanh1 sinh 2

gda + gd b = 2 tan1(ea + eb

1 ea+b) pi gd1 a + gd1 b = 2i tanh1

(eia + eib

1 eia+ib) pii

gda gd b = 2 tan1(ea+b + 1eb ea

) pi gd1 a gd1 b = 2i tanh1

(eia+ib + 1eia eib

) pii

cga + cg b = 2 tanh1(ea + eb

1 ea+b)

coa + co b = 2 coa + b

2

cga cg b = 2 tanh1(ea eb1 ea+b

)coa co b = b a

28

gsa + gs b = tan1(e2a + e2b

1 e2a+2b) pi

2gs1 a + gs1 b = i tanh1

(e2ia + e2ib

1 e2ia+2ib) pii

2

gsa gs b = tan1(e2a+2b + 1e2b e2a

) pi

2gs1 a gs1 b = i tanh1

(e2ia+2ib + 1e2ia e2ib

) pii

2

gd =k=0

E2k2k+1

(2k + 1)!gd1 =

k=0

(1)kE2k2k+1(2k + 1)!

= 2k=0

(1)k tanh2k+1 2

2k + 1= 2

k=0

tan2k+1

22k + 1

=pi

2k=0

(2k)! sech2k+1 4k(n!)2(2k + 1)

cg = ln +k=1

(1)k+12(22k1 1)Bk2k2k(2k)!

=k=0

(1)k(2k)! csch2k+1 4k(n!)2(2k + 1)

co gd = gd cg = cg co(i) gd1 co = cg gd1 = i co cg cg = gd1 co gd co = gd cg gd1

= co gd i co = co(i gd1 co )

Inner transformation conversions

cosh = cos i cos = cosh i= i sinh i co i = sin co i = i sinh i co = sin co = tanh gd1 co i = i tan i gd1 co i = tanh gd1 co = i tan i gd1 co = coth cg = i cot i cg = coth cg i = i cot i cg i= sech gd1 i = sec gd = sech gd1 = sec gd i= i csch i co gd = csc co gd = i csch i co gd i = csc co gd i

sinh = i cosh i co i = i cos co i sin = cosh i co = cos co = i sin i = i sinh i

= i tanh gd1 i = tan gd = tanh gd1 = i tan gd1 i= i coth i co gd = cot co gd = coth i co gd1 i = i cot co gd1 i= i sech gd1 co i = i sec i gd1 co i = sech gd1 co = sec i gd1 co = csch cg = i csc i cg = i csch cg i = csc i cg i

tanh = cosh i co gd = cos co gd tan = i cosh i co gd i = i cos co gd1 i= i sinh gd1 i = sin gd = sinh gd1 = i sin gd i

= i tan i = i tanh i= coth i co i = i cot co i = i coth i co = cot co = sech cg = sec i cg = i sech cg i = i sec i cg i= i csch gd1 co i = csc i gd1 co i = csch gd1 co = i csc i gd1 co

coth = cosh cg = cos i cg cot = i cosh cg i = i cos i gd1 i= i sinh gd1 co i = sin i gd1 co i = sinh gd1 co = i sin i gd1 co = tanh i co i = i tan co i = i tanh i co = tan co

= i cot i = i coth i= sech i co gd = sec co gd = i sech i co gd i = i sec co gd i= i csch gd1 i = csc gd = csch gd1 = i csc gd i

29

sech = cosh gd1 i = cos gd sec = cosh gd1 = cos gd i= i sinh i co gd = sin co gd = i sinh i co gd i = sin co gd i= tanh cg = i tan i cg = tanh cg i = i tan i cg i= coth gd1 co i = i cot i gd1 co i = coth gd1 co = i cot i gd1 co

= sec i = sech i= i csch i co i = csc co i = i csch i co = csc co

csch = i cosh gd1 co i = i cos i gd1 co i csc = cosh gd1 co = cos i gd1 co = sinh cg = i sin i cg = i sinh cg i = sin i cg i= i tanh i co gd = tan co gd = tanh i co gd i = i tan co gd1 i= i coth gd1 i = cot gd = coth gd1 = i cot gd i= i sech i co i = i sec co i = sech i co = sec co

= i csc i = i csch i

Alternative inner transformation conversions

cosh = sinh gd1 co gs cg = tan co gs cg cos = sinh gd1 gs gd co = tan gs gd1 co = csch gd1 gs cg = cot gs cg = csch gd1 co gs gd1 co = cot co gs gd1 co

sinh = cosh cg gs1 co gd = cos co gd gs1 gd sin = sinh gd1 gs gd = tan gs gd1 = tanh gs1 gd = sin gd1 gs1 gd = csch gd1 co gs gd1 = cot gs cg = coth gs1 co gd = sec co gd gs1 co gd = sech cg gs1 gd = csc gd gs1 co gd

tanh = sinh gd1 gs = tan gs tan = cosh cg gs1 co = cos co gd gs1 = csch gd1 co gs = cot co gs = tanh gs1 = sin gd gs1

= coth gs1 co = sec co gd gs1 co = sech cg gs1 = csc gd gs1 co

coth = sinh gd1 co gs = tan co gs cot = cosh cg gs1 = cos co gd gs1 co = csch gd1 gs = cot gs = tanh gs1 co = sin gd gs1 co

= coth gs1 = sec co gd gs1 = sech cg gs1 co = csc gd gs1

sech = sinh gd1 gs cg = tan gs cg sec = sinh gd1 co gs gd1 co = tan co gs gd1 co = csch gd1 co gs cg = cot co gs cg = csch gd1 gs gd1 co = cot gs gd1 co

csch = cosh cg gs1 gd = cos co gd gs1 co gd csc = sinh gd1 co gs gd1 = tan co gs gd1 = tanh gs1 co gd = sin gd gs1 co gd = csch gd1 gs gd1 = cot gs gd1 = coth gs1 gd = sec co gd gs1 gd = sech cg gs1 co gd = csc gd gs1 gd

DeMoivre identites

expn = exp(n) cisn = cis(n)1

expn = exp(n) 1

cisn = cis(n)

n

exp = exp

nn

cis = cis

nexpm+in = exp(m) cis(n) cism+in = cis(m) exp(n)

(exp t cis )m+in = exp ((m n)t) cis ((m + n))

30

Point on the complex plane

a + bi = et+i = exp t cis

= cosh t cos + sinh t cos + i cosh t sin + i sinh t sin

m

a = exp t cos = cosh t cos + sinh t cos

b = exp t sin = cosh t sin + sinh t sin

m

t = exp1a2 + b2 = exp1

(a + bi)(a bi) = tanh1 a

2 + b2 1a2 + b2 + 1

= cis1a + bia2 + b2

= cis1a + bia bi = tan

1 ba

m

cosh t =a2 + b2 + 1a2 + b2

cos =a

a2 + b2

sinh t =a2 + b2 1a2 + b2

sin =b

a2 + b2

Twice-applied formulae

ee

= exp exp = (cosh cosh + sinh cosh )(cosh sinh + sinh sinh )

eei

= exp cis = (cosh cos + sinh cos )(cos sin + i sin sin )

eie

= cis exp = (cos cosh + sin sinh )(cos sinh + i sin sinh )

eiei

= cis cis = (cos cos + i sin cos )(cosh sin sinh sin )

31

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