# §4.23 Inverse Trigonometric Functions

## §4.23(i) General Definitions

The general values of the inverse trigonometric functions are defined by

 4.23.1 $\displaystyle\operatorname{Arcsin}z$ $\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{(1-t^{2})^{1/2}},$ ⓘ Defines: $\operatorname{Arcsin}\NVar{z}$: general arcsine function Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $z$: complex variable A&S Ref: 4.4.1 (modified) Referenced by: §4.23(i), §4.23(ii), §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E1 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4 4.23.2 $\displaystyle\operatorname{Arccos}z$ $\displaystyle=\int_{z}^{1}\frac{\mathrm{d}t}{(1-t^{2})^{1/2}},$ ⓘ Defines: $\operatorname{Arccos}\NVar{z}$: general arccosine function Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $z$: complex variable A&S Ref: 4.4.2 (modified) Referenced by: §4.23(i) Permalink: http://dlmf.nist.gov/4.23.E2 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4 4.23.3 $\displaystyle\operatorname{Arctan}z$ $\displaystyle=\int_{0}^{z}\frac{\mathrm{d}t}{1+t^{2}},$ $z\neq\pm\mathrm{i}$, ⓘ Defines: $\operatorname{Arctan}\NVar{z}$: general arctangent function Symbols: $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable A&S Ref: 4.4.3 (modified) Referenced by: §4.23(i), §4.23(ii) Permalink: http://dlmf.nist.gov/4.23.E3 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4 4.23.4 $\displaystyle\operatorname{Arccsc}z$ $\displaystyle=\operatorname{Arcsin}\left(1/z\right),$ ⓘ Defines: $\operatorname{Arccsc}\NVar{z}$: general arccosecant function Symbols: $\operatorname{Arcsin}\NVar{z}$: general arcsine function and $z$: complex variable A&S Ref: 4.4.6 (modified) Permalink: http://dlmf.nist.gov/4.23.E4 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4 4.23.5 $\displaystyle\operatorname{Arcsec}z$ $\displaystyle=\operatorname{Arccos}\left(1/z\right),$ ⓘ Defines: $\operatorname{Arcsec}\NVar{z}$: general arcsecant function Symbols: $\operatorname{Arccos}\NVar{z}$: general arccosine function and $z$: complex variable A&S Ref: 4.4.7 (modified) Permalink: http://dlmf.nist.gov/4.23.E5 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4 4.23.6 $\displaystyle\operatorname{Arccot}z$ $\displaystyle=\operatorname{Arctan}\left(1/z\right).$ ⓘ Defines: $\operatorname{Arccot}\NVar{z}$: general arccotangent function Symbols: $\operatorname{Arctan}\NVar{z}$: general arctangent function and $z$: complex variable A&S Ref: 4.4.8 (modified) Permalink: http://dlmf.nist.gov/4.23.E6 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4

In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points $t=\pm 1$. The function $(1-t^{2})^{1/2}$ assumes its principal value when $t\in(-1,1)$; elsewhere on the integration paths the branch is determined by continuity. In (4.23.3) the integration path may not intersect $\pm i$. Each of the six functions is a multivalued function of $z$. $\operatorname{Arctan}z$ and $\operatorname{Arccot}z$ have branch points at $z=\pm\mathrm{i}$; the other four functions have branch points at $z=\pm 1$.

## §4.23(ii) Principal Values

The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. Compare the principal value of the logarithm (§4.2(i)). The principal branches are denoted by $\operatorname{arcsin}z$, $\operatorname{arccos}z$, $\operatorname{arctan}z$, respectively. Each is two-valued on the corresponding cuts, and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts.

The principal values of the inverse cosecant, secant, and cotangent are given by

 4.23.7 $\displaystyle\operatorname{arccsc}z$ $\displaystyle=\operatorname{arcsin}\left(1/z\right),$ ⓘ Defines: $\operatorname{arccsc}\NVar{z}$: arccosecant function Symbols: $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable Referenced by: §4.23(iv), §4.45(ii) Permalink: http://dlmf.nist.gov/4.23.E7 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4 4.23.8 $\displaystyle\operatorname{arcsec}z$ $\displaystyle=\operatorname{arccos}\left(1/z\right).$ ⓘ Defines: $\operatorname{arcsec}\NVar{z}$: arcsecant function Symbols: $\operatorname{arccos}\NVar{z}$: arccosine function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.23.E8 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4 4.23.9 $\displaystyle\operatorname{arccot}z$ $\displaystyle=\operatorname{arctan}\left(1/z\right),$ $z\neq\pm\mathrm{i}$. ⓘ Defines: $\operatorname{arccot}\NVar{z}$: arccotangent function Symbols: $\mathrm{i}$: imaginary unit, $\operatorname{arctan}\NVar{z}$: arctangent function and $z$: complex variable Referenced by: §4.23(iv), §4.45(ii) Permalink: http://dlmf.nist.gov/4.23.E9 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4

These functions are analytic in the cut plane depicted in Figures 4.23.1(iii) and 4.23.1(iv).

Except where indicated otherwise, it is assumed throughout the DLMF that the inverse trigonometric functions assume their principal values. Figure 4.23.1: z-plane. Branch cuts for the inverse trigonometric functions. Magnify

Graphs of the principal values for real arguments are given in §4.15. This section also includes conformal mappings, and surface plots for complex arguments.

## §4.23(iii) Reflection Formulas

 4.23.10 $\displaystyle\operatorname{arcsin}\left(-z\right)$ $\displaystyle=-\operatorname{arcsin}z,$ ⓘ Symbols: $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.14 Permalink: http://dlmf.nist.gov/4.23.E10 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.11 $\displaystyle\operatorname{arccos}\left(-z\right)$ $\displaystyle=\pi-\operatorname{arccos}z.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arccos}\NVar{z}$: arccosine function and $z$: complex variable A&S Ref: 4.4.15 Permalink: http://dlmf.nist.gov/4.23.E11 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.12 $\displaystyle\operatorname{arctan}\left(-z\right)$ $\displaystyle=-\operatorname{arctan}z,$ $z\neq\pm\mathrm{i}$. ⓘ Symbols: $\mathrm{i}$: imaginary unit, $\operatorname{arctan}\NVar{z}$: arctangent function and $z$: complex variable A&S Ref: 4.4.16 Permalink: http://dlmf.nist.gov/4.23.E12 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.13 $\displaystyle\operatorname{arccsc}\left(-z\right)$ $\displaystyle=-\operatorname{arccsc}z,$ ⓘ Symbols: $\operatorname{arccsc}\NVar{z}$: arccosecant function and $z$: complex variable A&S Ref: 4.4.17 Permalink: http://dlmf.nist.gov/4.23.E13 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.14 $\displaystyle\operatorname{arcsec}\left(-z\right)$ $\displaystyle=\pi-\operatorname{arcsec}z.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arcsec}\NVar{z}$: arcsecant function and $z$: complex variable A&S Ref: 4.4.18 Permalink: http://dlmf.nist.gov/4.23.E14 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.15 $\displaystyle\operatorname{arccot}\left(-z\right)$ $\displaystyle=-\operatorname{arccot}z,$ $z\neq\pm\mathrm{i}$. ⓘ Symbols: $\mathrm{i}$: imaginary unit, $\operatorname{arccot}\NVar{z}$: arccotangent function and $z$: complex variable A&S Ref: 4.4.19 (Early printings had an error.) Permalink: http://dlmf.nist.gov/4.23.E15 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.16 $\displaystyle\operatorname{arccos}z$ $\displaystyle=\tfrac{1}{2}\pi-\operatorname{arcsin}z,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arccos}\NVar{z}$: arccosine function, $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.2 Referenced by: §4.15(iii), §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E16 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.17 $\displaystyle\operatorname{arcsec}z$ $\displaystyle=\tfrac{1}{2}\pi-\operatorname{arccsc}z.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arccsc}\NVar{z}$: arccosecant function, $\operatorname{arcsec}\NVar{z}$: arcsecant function and $z$: complex variable A&S Ref: 4.4.9 Permalink: http://dlmf.nist.gov/4.23.E17 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4 4.23.18 $\displaystyle\operatorname{arccot}z$ $\displaystyle=\pm\tfrac{1}{2}\pi-\operatorname{arctan}z,$ $\Re z\gtrless 0$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arccot}\NVar{z}$: arccotangent function, $\operatorname{arctan}\NVar{z}$: arctangent function, $\Re$: real part and $z$: complex variable A&S Ref: 4.4.5 (The first printing had an error.) Referenced by: §4.15(iii) Permalink: http://dlmf.nist.gov/4.23.E18 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4

## §4.23(iv) Logarithmic Forms

Throughout this subsection all quantities assume their principal values.

### Inverse Sine

 4.23.19 $\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;

compare Figure 4.23.1(i). On the cuts

 4.23.20 $\displaystyle\operatorname{arcsin}x$ $\displaystyle=\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}+x\right),$ $x\in[1,\infty)$, 4.23.21 $\displaystyle\operatorname{arcsin}x$ $\displaystyle=-\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}-x\right),$ $x\in(-\infty,-1]$,

upper signs being taken on upper sides, and lower signs on lower sides.

### Inverse Cosine

 4.23.22 $\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;

compare Figure 4.23.1(i). An equivalent definition is

 4.23.23 $\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z}{2}\right)^{1/2}+i\left(% \frac{1-z}{2}\right)^{1/2}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;

see Kahan (1987).

On the cuts

 4.23.24 $\displaystyle\operatorname{arccos}x$ $\displaystyle=\mp i\ln\left((x^{2}-1)^{1/2}+x\right),$ $x\in[1,\infty)$, 4.23.25 $\displaystyle\operatorname{arccos}x$ $\displaystyle=\pi\mp i\ln\left((x^{2}-1)^{1/2}-x\right),$ $x\in(-\infty,-1]$,

the upper/lower signs corresponding to the upper/lower sides.

### Inverse Tangent

 4.23.26 $\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;

compare Figure 4.23.1(ii). On the cuts

 4.23.27 $\operatorname{arctan}\left(iy\right)=\pm\frac{1}{2}\pi+\frac{i}{2}\ln\left(% \frac{y+1}{y-1}\right),$ $y\in(-\infty,-1)\cup(1,\infty)$,

the upper/lower sign corresponding to the right/left side.

### Other Inverse Functions

For the corresponding results for $\operatorname{arccsc}z$, $\operatorname{arcsec}z$, and $\operatorname{arccot}z$, use (4.23.7)–(4.23.9). Care needs to be taken on the cuts, for example, if $0 then $1/(x+i0)=(1/x)-i0$.

## §4.23(v) Fundamental Property

With $k\in\mathbb{Z}$, the general solutions of the equations

 4.23.28 $\displaystyle z$ $\displaystyle=\sin w,$ ⓘ Symbols: $\sin\NVar{z}$: sine function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.23.E28 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4 4.23.29 $\displaystyle z$ $\displaystyle=\cos w,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.23.E29 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4 4.23.30 $\displaystyle z$ $\displaystyle=\tan w,$ ⓘ Symbols: $\tan\NVar{z}$: tangent function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.23.E30 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4

are respectively

 4.23.31 $\displaystyle w$ $\displaystyle=\operatorname{Arcsin}z=(-1)^{k}\operatorname{arcsin}z+k\pi,$ 4.23.32 $\displaystyle w$ $\displaystyle=\operatorname{Arccos}z=\pm\operatorname{arccos}z+2k\pi,$ 4.23.33 $\displaystyle w$ $\displaystyle=\operatorname{Arctan}z=\operatorname{arctan}z+k\pi,$ $z\neq\pm\mathrm{i}$.

## §4.23(vi) Real and Imaginary Parts

 4.23.34 $\displaystyle\operatorname{arcsin}z$ $\displaystyle=\operatorname{arcsin}\beta+\mathrm{i}\operatorname{sign}\left(y% \right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right),$ ⓘ Symbols: $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of, $\mathrm{i}$: imaginary unit, $\operatorname{arcsin}\NVar{z}$: arcsine function, $\ln\NVar{z}$: principal branch of logarithm function, $\notin$: not an element of, $(\NVar{a},\NVar{b})$: open interval, $\operatorname{sign}\NVar{x}$: sign of, $x$: real variable, $y$: real variable, $z$: complex variable, $\alpha$ and $\beta$ A&S Ref: 4.4.37 (with the general value.) Referenced by: §4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35) Permalink: http://dlmf.nist.gov/4.23.E34 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\operatorname{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the originally stated condition $x\in[-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin(1,\infty)$. Reported 2013-07-01 by Volker Thürey See also: Annotations for §4.23(vi), §4.23 and Ch.4 4.23.35 $\displaystyle\operatorname{arccos}z$ $\displaystyle=\operatorname{arccos}\beta-\mathrm{i}\operatorname{sign}\left(y% \right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right),$ ⓘ Symbols: $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of, $\mathrm{i}$: imaginary unit, $\operatorname{arccos}\NVar{z}$: arccosine function, $\ln\NVar{z}$: principal branch of logarithm function, $\notin$: not an element of, $(\NVar{a},\NVar{b})$: open interval, $\operatorname{sign}\NVar{x}$: sign of, $x$: real variable, $y$: real variable, $z$: complex variable, $\alpha$ and $\beta$ A&S Ref: 4.4.38 (with the general value.) Referenced by: §4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35) Permalink: http://dlmf.nist.gov/4.23.E35 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\operatorname{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the originally stated condition $x\in[-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin(1,\infty)$. Reported 2013-07-01 by Volker Thürey See also: Annotations for §4.23(vi), §4.23 and Ch.4 4.23.36 $\displaystyle\operatorname{arctan}z$ $\displaystyle=\tfrac{1}{2}\operatorname{arctan}\left(\frac{2x}{1-x^{2}-y^{2}}% \right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}\right),$ ⓘ Symbols: $\mathrm{i}$: imaginary unit, $\operatorname{arctan}\NVar{z}$: arctangent function, $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable, $y$: real variable and $z$: complex variable A&S Ref: 4.4.39 (with the general value.) Referenced by: (4.23.34), (4.23.35), §4.23(vi) Permalink: http://dlmf.nist.gov/4.23.E36 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4

where $z=x+\mathrm{i}y$ and $\pm z\notin(1,\infty)$ in (4.23.34) and (4.23.35), and $\left|z\right|<1$ in (4.23.36). Also,

 4.23.37 $\displaystyle\alpha$ $\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}+\tfrac{1}{2}\left% ((x-1)^{2}+y^{2}\right)^{1/2},$ ⓘ Defines: $\alpha$ (locally) Symbols: $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/4.23.E37 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4 4.23.38 $\displaystyle\beta$ $\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}-\tfrac{1}{2}\left% ((x-1)^{2}+y^{2}\right)^{1/2}.$ ⓘ Defines: $\beta$ (locally) Symbols: $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/4.23.E38 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4

## §4.23(vii) Special Values and Interrelations

For interrelations see Table 4.16.3. For example, from the heading and last entry in the penultimate column we have $\operatorname{arcsec}a=\operatorname{arccot}\left((a^{2}-1)^{-1/2}\right)$.

## §4.23(viii) Gudermannian Function

The Gudermannian $\operatorname{gd}\left(x\right)$ is defined by

 4.23.39 $\operatorname{gd}\left(x\right)=\int_{0}^{x}\operatorname{sech}t\mathrm{d}t,$ $-\infty. ⓘ Defines: $\operatorname{gd}\NVar{x}$: Gudermannian function Symbols: $\mathrm{d}\NVar{x}$: differential, $\operatorname{sech}\NVar{z}$: hyperbolic secant function, $\int$: integral and $x$: real variable Referenced by: §4.40(ii) Permalink: http://dlmf.nist.gov/4.23.E39 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

Equivalently,

 4.23.40 $\operatorname{gd}\left(x\right)=2\operatorname{arctan}\left(e^{x}\right)-% \tfrac{1}{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x% \right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}% \left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(% \operatorname{csch}x\right).$

The inverse Gudermannian function is given by

 4.23.41 ${\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x}\sec t\mathrm{d}t,$ $-\frac{1}{2}\pi. ⓘ Defines: ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$: inverse Gudermannian function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\sec\NVar{z}$: secant function and $x$: real variable Referenced by: §19.9(ii), §4.26(ii) Permalink: http://dlmf.nist.gov/4.23.E41 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

Equivalently, and again when $-\frac{1}{2}\pi,

 4.23.42 ${\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}% \pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x% \right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(% \sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}% \left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).$