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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 . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.7Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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
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The unit circle
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tan |
cot
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/2x
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
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The unit hyperbola
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cosh
||||
sinh ||||
||tanh ||
coth
csch sech
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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
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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
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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
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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
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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
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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))
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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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