Nodiant
-------

Defnyddir y nodiant canlynol ar gyfer pob un o'r chwech ffwythiant trigonometrig (sin, cosin (cos), tangiad (tan), cotangiad (cot), secant (sec), a chosecant (csc). Dim ond y nodiant ar gyfer sin a roddir isod, mae'r nodiant ar gyfer y ffwythiannau eraill yn gyffelyb.

| Nodiant | Darllener | Disgrifiad | Diffiniad |
| --- | --- | --- | --- |
| sin²(x) | "sin sgwâr x" | sin wedi ei sgwario | sin²(*x*) = (sin(x))² |
| arcsin(x) | "arcsin x" | ffwythiant gwrthdro sin | arcsin(x) = y  os a dim ond os  sin(y) = x a {\displaystyle -{\pi \over 2}\leq y\leq {\pi \over 2}}  |
| (sin(x))−1 | "sin x, i'r [pŵer] meinws un" | Cilydd sin | (sin(x))−1 = 1 / sin(*x*) = csc(*x*) |

Gellir ysgrifennu arcsin(*x*) yn sin−1(*x*) yn ogystal; rhaid gofalu rhag drysu hyn â (sin(*x*))−1.

Diffiniadau
-----------

{\displaystyle {\begin{aligned}\cos(x)&=\sin \left(x+{\frac {\pi }{2}}\right)\\\tan(x)&={\frac {\sin(x)}{\cos(x)}}&\quad \cot(x)&={\frac {\cos(x)}{\sin(x)}}={\frac {1}{\tan(x)}}\\\sec(x)&={\frac {1}{\cos(x)}}&\quad \csc(x)&={\frac {1}{\sin(x)}}\end{aligned}}}
(Gweler ffwythiant trigonometrig am fwy o wybodaeth)

Cyfnodedd, cymesuredd a symudiadau
----------------------------------

### Cyfnodedd

Mae cyfnod o 2π gan y ffwythiannau sin, cosin, secant, a chosecant (cylch llawn): os mae {\displaystyle k} yn unrhyw gyfanrif yna mae

{\displaystyle {\begin{aligned}\sin(x)&=\sin(x+2k\pi )\\\cos(x)&=\cos(x+2k\pi )\\\sec(x)&=\sec(x+2k\pi )\\\csc(x)&=\csc(x+2k\pi )\\\end{aligned}}}
Mae cyfnod o π (hanner cylch) gan y ffwythiannau tangiad a chotangiad:

{\displaystyle {\begin{aligned}\tan(x)&=\tan(x+k\pi )\\\cot(x)&=\cot(x+k\pi )\\\end{aligned}}}
### Cymesuredd

{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)&\sin \left({\tfrac {\pi }{2}}-x\right)&=\cos(x)&\sin \left(\pi -x\right)&=+\sin(x)\\\cos(-x)&=+\cos(x)&\cos \left({\tfrac {\pi }{2}}-x\right)&=\sin(x)&\cos \left(\pi -x\right)&=-\cos(x)\\\tan(-x)&=-\tan(x)&\tan \left({\tfrac {\pi }{2}}-x\right)&=\cot(x)&\tan \left(\pi -x\right)&=-\tan(x)\\\cot(-x)&=-\cot(x)&\cot \left({\tfrac {\pi }{2}}-x\right)&=\tan(x)&\cot \left(\pi -x\right)&=-\cot(x)\\\sec(-x)&=+\sec(x)&\sec \left({\tfrac {\pi }{2}}-x\right)&=\csc(x)&\sec \left(\pi -x\right)&=-\sec(x)\\\csc(-x)&=-\csc(x)&\csc \left({\tfrac {\pi }{2}}-x\right)&=\sec(x)&\csc \left(\pi -x\right)&=+\csc(x)\end{aligned}}}
### Symudiadau

{\displaystyle {\begin{aligned}\sin \left(x+{\tfrac {\pi }{2}}\right)&=+\cos(x)&\sin \left(x+\pi \right)&=-\sin(x)\\\cos \left(x+{\tfrac {\pi }{2}}\right)&=-\sin(x)&\cos \left(x+\pi \right)&=-\cos(x)\\\tan \left(x+{\tfrac {\pi }{2}}\right)&=-\cot(x)&\tan \left(x+\pi \right)&=+\tan(x)\\\cot \left(x+{\tfrac {\pi }{2}}\right)&=-\tan(x)&\cot \left(x+\pi \right)&=+\cot(x)\\\sec \left(x+{\tfrac {\pi }{2}}\right)&=-\csc(x)&\sec \left(x+\pi \right)&=-\sec(x)\\\csc \left(x+{\tfrac {\pi }{2}}\right)&=+\sec(x)&\csc \left(x+\pi \right)&=-\csc(x)\end{aligned}}}
### Cyfuniadau llinol

Weithiau mae'n bwysig gwybod bod cyfuniad llinol o donau sin gyda'r un cyfnod (ond gyda gwahanol symudiad cydwedd) yn rhoi ton sin gyda'r un cyfnod. Yn gyffrefinol, mae

{\displaystyle a\sin x+b\cos x={\sqrt {a^{2}+b^{2}}}\cdot \sin(x+\varphi )\,}
lle mae

{\displaystyle \varphi =\left\{{\begin{matrix}{\rm {arctan}}(b/a),&&{\mbox{os mae }}a\geq 0;\;\\\arctan(b/a)\pm \pi ,&&{\mbox{os mae }}a<0.\;\end{matrix}}\right.\;}
Yn gyffredinol, am symudiad cydwedd mympwyol, mae gennym fod

{\displaystyle a\sin x+b\sin(x+\alpha )=c\sin(x+\beta )\,}
lle mae

{\displaystyle c={\sqrt {a^{2}+b^{2}+2ab\cos \alpha }},}
a

{\displaystyle \beta ={\rm {arctan}}\left({\frac {b\sin \alpha }{a+b\cos \alpha }}\right).}

Unfathiannau Pythagoreaidd
--------------------------

Seilir y canlynol ar theorem Pythagoras:

{\displaystyle {\begin{aligned}\sin ^{2}(x)+\cos ^{2}(x)&=1\\\tan ^{2}(x)+1&=\sec ^{2}(x)\\\cot ^{2}(x)+1&=\csc ^{2}(x)\end{aligned}}}
Gellir deillio'r ail a'r trydydd hafaliad uchod o'r cyntaf trwy rhannu â cos2(*x*) a sin2(*x*) yn ôl eu trefn.

Unfathiannau swm neu wahaniaeth onglau
--------------------------------------

Fe'u celwir hefyd yn "fformwlâu adio a thynnu". Gellir eu profi gan ddefnyddio fformwla Euler.

{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}
(Pan y mae "+" ar y chwith, mae "+" ar y de, ac yn gyffelyb gyda "-".)
{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}
(Pan y mae "+" ar y chwith, mae "-" ar y de, ac i'r gwrthwyneb.)
{\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}}
### Tangiad symiau nifer meidraidd o dermau

Gadewch i *x**i* = tan(θ*i* ), ar gyfer *i* = 1, ... *n*. Gadewch i *e**k* fod y polynomial cymesur elfennol gyda gradd *k* yn y newidynnau *x**i*, *i* = 1, ... *n*, *k* = 0, ... *n*. Yna mae

{\displaystyle \tan(\theta \_{1}+\cdots +\theta \_{n})={\frac {e\_{1}-e\_{3}+e\_{5}-\cdots }{e\_{0}-e\_{2}+e\_{4}-\cdots }},}
gyda'r nifer o dermau yn dibynnu ar *n*.

Er enghraifft, mae

{\displaystyle {\begin{aligned}\tan(\theta \_{1}+\theta \_{2}+\theta \_{3})&{}={\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}}}={\frac {(x\_{1}+x\_{2}+x\_{3})\ -\ (x\_{1}x\_{2}x\_{3})}{1\ -\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{2}x\_{3})}},\\\\\tan(\theta \_{1}+\theta \_{2}+\theta \_{3}+\theta \_{4})&{}={\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}+e\_{4}}}\\\\&{}={\frac {(x\_{1}+x\_{2}+x\_{3}+x\_{4})\ -\ (x\_{1}x\_{2}x\_{3}+x\_{1}x\_{2}x\_{4}+x\_{1}x\_{3}x\_{4}+x\_{2}x\_{3}x\_{4})}{1\ -\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{1}x\_{4}+x\_{2}x\_{3}+x\_{2}x\_{4}+x\_{3}x\_{4})\ +\ (x\_{1}x\_{2}x\_{3}x\_{4})}},\end{aligned}}}
ac yn y blaen. Gellir profi hyn trwy anwythiad mathemategol.

Fformwlâu ongl dwbl
-------------------

Gellir profi'r canlynol trwy amnewid *x* = *y* yn y fformwlâu adio, a defnyddio'r fformwla Pythagoreaidd, neu trwy ddefnyddio fformwla de Moivre gydag *n* = 2.

{\displaystyle \sin(2x)=2\sin(x)\cos(x)\,}
{\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)={\frac {1-\tan ^{2}(x)}{1+\tan ^{2}(x)}}\,}
{\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}\,}

  

{\displaystyle \cot(2x)={\frac {\cot(x)-\tan(x)}{2}}\,}
Gellir defnyddio'r uchod i ganfod triawdau Pythagoraidd. os mae (*a*, *b*, *c*) yw hyd ochrau triongl ongl-sgwâr, yna mae (*a*2 − *b*2, 2*ab*, *c*2) hefyd yn ffurfio triongl ongl-sgwâr, lle mae *B* yw'r ongl a ddyblir. os mae *a*2 − *b*2 yn negatif, cymerwch ei wrthdro a defnyddio ongl cyflenwol 2*B* yn lle 2*B*.

Fformwlâu ongl triphlyg
-----------------------

{\displaystyle \sin(3x)=3\sin(x)-4\sin ^{3}(x)\,}
{\displaystyle \cos(3x)=4\cos ^{3}(x)-3\cos(x)\,}
{\displaystyle \tan(3x)={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}(x)}}}

Fformwlâu aml-ongl
------------------

Os mai *Tn* yw'r *n*fed polynomial Chebyshev, yna mae

{\displaystyle \cos(nx)=T\_{n}(\cos(x)).\,}
Os mai *S**n* yw'r *n*fed polynomial gwasgar, yna mae

{\displaystyle \sin ^{2}(n\theta )=S\_{n}(\sin ^{2}\theta ).\,}
Fformwla de Moivre:

{\displaystyle \cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^{n}\,}

Fformwlâu lleihau pŵer
----------------------

{\displaystyle \sin ^{2}(x)={1-\cos(2x) \over 2}}
{\displaystyle \cos ^{2}(x)={1+\cos(2x) \over 2}}
{\displaystyle \sin ^{2}(x)\cos ^{2}(x)={1-\cos(4x) \over 8}}
{\displaystyle \sin ^{3}(x)={\frac {3\sin(x)-\sin(3x)}{4}}}
{\displaystyle \cos ^{3}(x)={\frac {3\cos(x)+\cos(3x)}{4}}}

Fformwlâu hanner ongl
---------------------

{\displaystyle \cos \left({\frac {x}{2}}\right)=\pm \,{\sqrt {\frac {1+\cos(x)}{2}}}}
{\displaystyle \sin \left({\frac {x}{2}}\right)=\pm \,{\sqrt {\frac {1-\cos(x)}{2}}}}
{\displaystyle \tan \left({\frac {x}{2}}\right)={\sin(x/2) \over \cos(x/2)}=\pm \,{\sqrt {1-\cos x \over 1+\cos x}}.\qquad \qquad (1)}
{\displaystyle \tan \left({\frac {x}{2}}\right)=\pm \,{\sqrt {(1-\cos x)(1+\cos x) \over (1+\cos x)(1+\cos x)}}=\pm \,{\sqrt {1-\cos ^{2}x \over (1+\cos x)^{2}}}}
{\displaystyle ={\sin x \over 1+\cos x}.}
{\displaystyle \tan \left({\frac {x}{2}}\right)=\pm \,{\sqrt {(1-\cos x)(1-\cos x) \over (1+\cos x)(1-\cos x)}}=\pm \,{\sqrt {(1-\cos x)^{2} \over (1-\cos ^{2}x)}}}
{\displaystyle ={1-\cos x \over \sin x}.}
{\displaystyle \tan \left({\frac {x}{2}}\right)={\frac {\sin(x)}{1+\cos(x)}}={\frac {1-\cos(x)}{\sin(x)}}.}
{\displaystyle \tan \left({x \over 2}\right)=\csc(x)-\cot(x),}
{\displaystyle \cot \left({x \over 2}\right)=\csc(x)+\cot(x).}
{\displaystyle t=\tan \left({\frac {x}{2}}\right),}

|  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
|     | {\displaystyle \sin(x)={\frac {2t}{1+t^{2}}}} |   a   | {\displaystyle \cos(x)={\frac {1-t^{2}}{1+t^{2}}}} |   a   | {\displaystyle e^{ix}={\frac {1+it}{1-it}}.} |

Amnewidiad o *t* am tan(*x*/2) yw hyn, gyda'r canlyniad fod sin(*x*) yn newid yn 2*t*/(1 + *t*2) a cos(*x*) yn (1 − *t*2)/(1 + *t*2). Mae hyn yn ddefnyddiol mewn calcwlws ar gyfer integreiddio ffwythiannau cymarebol o sin(*x*) a cos(*x*).

Unfathiannau lluoswm-i-swm
--------------------------

{\displaystyle \cos \left(x\right)\cos \left(y\right)={\cos \left(x-y\right)+\cos \left(x+y\right) \over 2}\;}
{\displaystyle \sin \left(x\right)\sin \left(y\right)={\cos \left(x-y\right)-\cos \left(x+y\right) \over 2}\;}
{\displaystyle \sin \left(x\right)\cos \left(y\right)={\sin \left(x-y\right)+\sin \left(x+y\right) \over 2}\;}
(gw. Theorem Ptolemi)

Unfathiannau swm-i-lluoswm
--------------------------

{\displaystyle \cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\;}
{\displaystyle \sin(x)+\sin(y)=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\;}
{\displaystyle \cos(x)-\cos(y)=-2\sin \left({x+y \over 2}\right)\sin \left({x-y \over 2}\right)\;}
{\displaystyle \sin(x)-\sin(y)=2\cos \left({x+y \over 2}\right)\sin \left({x-y \over 2}\right)\;}
fformwla de Moivre

{\displaystyle {\mbox{os mae }}x+y+z=\pi ,}
{\displaystyle {\mbox{yna mae }}\tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z).\,}
(Os am roi ystyr i'r fformwla tra fod unrhyw un o *x*, *y*, a *z* yn ongl sgwâr, rhaid cymryd mai ∞ yw'r ddau ochr. Nid +∞ neu −∞ yw hyn, ond un pwynt "at anfeidredd" a ychwanegir i'r linell rif real.)

{\displaystyle {\mbox{Os mae }}x+y+z=\pi ={\mbox{hanner cylch,}}\,}
{\displaystyle {\mbox{yna mae }}\sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z).\,}

Ffwythiannau trigonometrig gwrthdro
-----------------------------------

{\displaystyle \arcsin(x)+\arccos(x)=\pi /2\;}
{\displaystyle \arctan(x)+\operatorname {arccot}(x)=\pi /2.\;}
{\displaystyle \arctan(x)+\arctan(1/x)=\left\{{\begin{matrix}\pi /2,&{\mbox{Os mae }}x>0\\-\pi /2,&{\mbox{os mae }}x<0\end{matrix}}\right.}
{\displaystyle \arctan(x)+\arctan(y)=\arctan \left({\frac {x+y}{1-xy}}\right)+\left\{{\begin{matrix}\pi ,&{\mbox{os mae }}x,y>0\\-\pi ,&{\mbox{os mae }}x,y<0\\0,&{\mbox{fel arall }}\end{matrix}}\right.}
{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}
{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}
{\displaystyle \sin[\arctan(x)]={\frac {x}{\sqrt {1+x^{2}}}}}
{\displaystyle \cos[\arctan(x)]={\frac {1}{\sqrt {1+x^{2}}}}}
{\displaystyle \tan[\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
{\displaystyle \tan[\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}

Perthynas gyda'r ffwythiant esbonyddol cymhlyg
----------------------------------------------

{\displaystyle e^{ix}=\cos(x)+i\sin(x)\,}
{\displaystyle e^{-ix}=\cos(x)-i\sin(x)\,}
{\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}\;}
{\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}\;}
lle mae *i*2 = −1.

Gw. fformwla Euler.

Diffiniadau esbonyddol
----------------------

{\displaystyle \sin(\theta )={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}
{\displaystyle \cos(\theta )={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}
{\displaystyle \tan(\theta )={\frac {\sin(\theta )}{\operatorname {cosh} (\theta )}}={\frac {({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}\,}
{\displaystyle \cot(\theta )={\frac {\cos(\theta )}{\sin(\theta )}}={\frac {({\frac {e^{i\theta }+e^{-i\theta }}{2}})}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}\,}
{\displaystyle \sec(\theta )={\frac {1}{\cos(\theta )}}={\frac {1}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}\,}
{\displaystyle \csc(\theta )={\frac {1}{\sin(\theta )}}={\frac {1}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}\,}
{\displaystyle \operatorname {versin} (\theta )=1-\cos(\theta )=1-{\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}
{\displaystyle \operatorname {vercos} (\theta )=1-\sin(\theta )=1-{\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}
{\displaystyle \operatorname {exsec} (\theta )=\operatorname {sec} (\theta )-1\ ={\frac {1}{\cos(\theta )}}-1={\frac {1}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}-1\,}
{\displaystyle \operatorname {excsc} (\theta )=\operatorname {csc} (\theta )-1\ ={\frac {1}{\sin(\theta )}}-1={\frac {1}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}-1\,}
{\displaystyle \operatorname {sinh} (\theta )=-i\sin(i\theta )={\frac {e^{\theta }-e^{-\theta }}{2}}\,}
{\displaystyle \operatorname {cosh} (\theta )=\cos(i\theta )={\frac {e^{\theta }+e^{-\theta }}{2}}\,}
{\displaystyle \operatorname {tanh} (\theta )=-i\tan(i\theta )={\frac {\operatorname {sinh} (\theta )}{\operatorname {cosh} (\theta )}}={\frac {e^{\theta }-e^{-\theta }}{e^{\theta }+e^{-\theta }}}={\frac {e^{2\theta }-1}{e^{2\theta }+1}}\,}
{\displaystyle \operatorname {coth} (\theta )=i\operatorname {cot} (i\theta )={\frac {\operatorname {cosh} (\theta )}{\operatorname {sinh} (\theta )}}={\frac {e^{\theta }+e^{-\theta }}{e^{\theta }-e^{-\theta }}}={\frac {e^{2\theta }+1}{e^{2\theta }-1}}\,}
{\displaystyle \operatorname {sech} (\theta )={\frac {1}{\operatorname {cosh} (\theta )}}=\operatorname {sec} (i\theta )={\frac {2}{e^{\theta }+e^{-\theta }}}\,}
{\displaystyle \operatorname {csch} (\theta )={\frac {1}{\operatorname {sinh} (\theta )}}=i\cos(i\theta )={\frac {2}{e^{\theta }-e^{-\theta }}}\,}
{\displaystyle \operatorname {versinh} (\theta )=1-\cos(i\theta )=1-{\frac {e^{\theta }+e^{-\theta }}{2}}\,}
{\displaystyle \operatorname {vercosh} (\theta )=1+i\sin(i\theta )=1-{\frac {e^{\theta }-e^{-\theta }}{2}}\,}
{\displaystyle \operatorname {exsech} (\theta )=\operatorname {sech} (\theta )-1={\frac {1}{\operatorname {cosh} (\theta )}}-1=\operatorname {sec} (i\theta )={\frac {2}{e^{\theta }+e^{-\theta }}}-1\,}
{\displaystyle \operatorname {excsch} (\theta )=\operatorname {csch} (\theta )-1={\frac {1}{\operatorname {sinh} (\theta )}}-1=i\cos(i\theta )={\frac {2}{e^{\theta }-e^{-\theta }}}-1\,}
{\displaystyle \arcsin(\theta )=-i\ln(i\theta +{\sqrt {1-\theta ^{2}}})\,}
{\displaystyle \arccos(\theta )=-i\ln(\theta +i{\sqrt {1-\theta ^{2}}})\,}
{\displaystyle \arctan(\theta )={\frac {\ln({\frac {i+\theta }{i-\theta }})i}{2}}\,}
{\displaystyle \operatorname {arccot}(\theta )=\arctan(-\theta )={\frac {i\ln({\frac {i-\theta }{i+\theta }})}{2}}\,}
{\displaystyle \operatorname {arcsec}(\theta )=\arccos(\theta ^{-1})=-i\ln(\theta ^{-1}+{\sqrt {1-\theta ^{-2}}}i)\,}
{\displaystyle \operatorname {arccsc}(\theta )=\arcsin(\theta ^{-1})=-i\ln(i\theta ^{-1}+{\sqrt {1-\theta ^{-2}}})\,}
{\displaystyle \operatorname {arcversin} (\theta )=\arccos(1-\theta )=-i\ln(1-\theta +i{\sqrt {1-(1-\theta )^{2}}})\,}
{\displaystyle \operatorname {arcvercos} (\theta )=\operatorname {arcsin} (1-\theta )=-i\ln(i-i\theta +{\sqrt {1-(1-\theta )^{2}}})\,}
{\displaystyle \operatorname {arcexsec} (\theta )=\operatorname {arcsec}(1+\theta )=-i\ln((\theta +1)^{-1}+i{\sqrt {1-(1+\theta )^{2}}})\,}
{\displaystyle \operatorname {arcexcsc} (\theta )=\operatorname {arccsc}(1+\theta )=-i\ln(i(\theta +1)^{-1}+{\sqrt {1-(1+\theta )^{2}}})\,}
{\displaystyle \operatorname {arcsinh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}+1}})\,}
{\displaystyle \operatorname {arccosh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}-1}})\,}
{\displaystyle \operatorname {arctanh} (\theta )={\frac {\ln({\frac {i+\theta }{i-\theta }})}{2}}\,}
{\displaystyle \operatorname {arccoth} (\theta )=\operatorname {arctanh} (-\theta )={\frac {\ln({\frac {i-\theta }{i+\theta }})}{2}}\,}
{\displaystyle \operatorname {arcsech} (\theta )=\operatorname {arccosh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}-1}})\,}
{\displaystyle \operatorname {arccsch} (\theta )=\operatorname {arcsinh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}+1}})\,}
{\displaystyle \operatorname {arcversinh} (\theta )=\operatorname {arccosh} (\theta )-1=\ln(\theta +{\sqrt {\theta ^{2}-1}})-1\,}
{\displaystyle \operatorname {arcvercosh} (\theta )=\operatorname {arcsinh} (\theta )-1=\ln(\theta +{\sqrt {\theta ^{2}+1}})-1\,}
{\displaystyle \operatorname {arcexsech} (\theta )=\operatorname {arcsech} (\theta +1)=\operatorname {arccosh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}-1}})\,}
{\displaystyle \operatorname {arcexcsch} (\theta )=\operatorname {arccsch} (\theta +1)=\operatorname {arcsinh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}+1}})\,}