# §4.24 Inverse Trigonometric Functions: Further Properties

## §4.24(i) Power Series

 4.24.1 $\operatorname{arcsin}z=z+\frac{1}{2}\frac{z^{3}}{3}+\frac{1\cdot 3}{2\cdot 4}% \frac{z^{5}}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^{7}}{7}+\cdots,$ $|z|\leq 1$. ⓘ Symbols: $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.40 (where the constraint is $|z|<1$.) Permalink: http://dlmf.nist.gov/4.24.E1 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4
 4.24.2 $\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),$ $|1-z|\leq 2$. ⓘ Symbols: $!$: factorial (as in $n!$), $\operatorname{arccos}\NVar{z}$: arccosine function, $n$: integer and $z$: complex variable A&S Ref: 4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.) Permalink: http://dlmf.nist.gov/4.24.E2 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4
 4.24.3 $\operatorname{arctan}z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots,$ $\left|z\right|\leq 1$, $z\neq\pm\mathrm{i}$. ⓘ Symbols: $\mathrm{i}$: imaginary unit, $\operatorname{arctan}\NVar{z}$: arctangent function and $z$: complex variable A&S Ref: 4.4.42 Referenced by: §3.10(ii), §4.45(i), §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E3 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4
 4.24.4 $\operatorname{arctan}z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{% 5z^{5}}+\cdots,$ $\Re z\gtrless 0$, $|z|\geq 1$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$: arctangent function, $\Re$: real part and $z$: complex variable A&S Ref: 4.4.42 (has an error. $\frac{\pi}{2}$ should have a negative sign when $\Re z<0$.) Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E4 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4
 4.24.5 $\operatorname{arctan}z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}\frac{z^{2}}{1+z^% {2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}\right)^{2}+\cdots% \right),$ $\Re\left(z^{2}\right)>-\tfrac{1}{2}$, ⓘ Symbols: $\operatorname{arctan}\NVar{z}$: arctangent function, $\Re$: real part and $z$: complex variable A&S Ref: 4.4.42 (has an error in the conditions on $z$.) Permalink: http://dlmf.nist.gov/4.24.E5 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4

which requires $z$ $(=x+iy)$ to lie between the two rectangular hyperbolas given by

 4.24.6 $x^{2}-y^{2}=-\tfrac{1}{2}.$ ⓘ Symbols: $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/4.24.E6 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4

## §4.24(ii) Derivatives

 4.24.7 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin}z$ $\displaystyle=(1-z^{2})^{-1/2},$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.52 Permalink: http://dlmf.nist.gov/4.24.E7 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4 4.24.8 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos}z$ $\displaystyle=-(1-z^{2})^{-1/2},$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arccos}\NVar{z}$: arccosine function and $z$: complex variable A&S Ref: 4.4.53 Permalink: http://dlmf.nist.gov/4.24.E8 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4 4.24.9 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan}z$ $\displaystyle=\frac{1}{1+z^{2}}.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arctan}\NVar{z}$: arctangent function and $z$: complex variable A&S Ref: 4.4.54 Permalink: http://dlmf.nist.gov/4.24.E9 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4 4.24.10 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z$ $\displaystyle=\mp\frac{1}{z(z^{2}-1)^{1/2}},$ $\Re z\gtrless 0$. ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arccsc}\NVar{z}$: arccosecant function, $\Re$: real part and $z$: complex variable A&S Ref: 4.4.57 (has an error.) Referenced by: §4.24(ii) Permalink: http://dlmf.nist.gov/4.24.E10 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4 4.24.11 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z$ $\displaystyle=\pm\frac{1}{z(z^{2}-1)^{1/2}},$ $\Re z\gtrless 0$. ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arcsec}\NVar{z}$: arcsecant function, $\Re$: real part and $z$: complex variable A&S Ref: 4.4.56 (has an error.) Referenced by: §4.24(ii) Permalink: http://dlmf.nist.gov/4.24.E11 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4 4.24.12 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot}z$ $\displaystyle=-\frac{1}{1+z^{2}}.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\operatorname{arccot}\NVar{z}$: arccotangent function and $z$: complex variable A&S Ref: 4.4.55 Permalink: http://dlmf.nist.gov/4.24.E12 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4

## §4.24(iii) Addition Formulas

 4.24.13 $\operatorname{Arcsin}u\pm\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1% -v^{2})^{1/2}\pm v(1-u^{2})^{1/2}\right),$ ⓘ Symbols: $\operatorname{Arcsin}\NVar{z}$: general arcsine function A&S Ref: 4.4.32 Permalink: http://dlmf.nist.gov/4.24.E13 Encodings: TeX, pMML, png See also: Annotations for §4.24(iii), §4.24 and Ch.4
 4.24.14 $\operatorname{Arccos}u\pm\operatorname{Arccos}v=\operatorname{Arccos}\left(uv% \mp((1-u^{2})(1-v^{2}))^{1/2}\right),$ ⓘ Symbols: $\operatorname{Arccos}\NVar{z}$: general arccosine function A&S Ref: 4.4.33 Permalink: http://dlmf.nist.gov/4.24.E14 Encodings: TeX, pMML, png See also: Annotations for §4.24(iii), §4.24 and Ch.4
 4.24.15 $\operatorname{Arctan}u\pm\operatorname{Arctan}v=\operatorname{Arctan}\left(% \frac{u\pm v}{1\mp uv}\right),$ ⓘ Symbols: $\operatorname{Arctan}\NVar{z}$: general arctangent function A&S Ref: 4.4.34 Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E15 Encodings: TeX, pMML, png See also: Annotations for §4.24(iii), §4.24 and Ch.4
 4.24.16 $\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),$ ⓘ Symbols: $\operatorname{Arccos}\NVar{z}$: general arccosine function and $\operatorname{Arcsin}\NVar{z}$: general arcsine function A&S Ref: 4.4.35 Permalink: http://dlmf.nist.gov/4.24.E16 Encodings: TeX, pMML, png See also: Annotations for §4.24(iii), §4.24 and Ch.4
 4.24.17 $\operatorname{Arctan}u\pm\operatorname{Arccot}v=\operatorname{Arctan}\left(% \frac{uv\pm 1}{v\mp u}\right)=\operatorname{Arccot}\left(\frac{v\mp u}{uv\pm 1% }\right).$ ⓘ Symbols: $\operatorname{Arccot}\NVar{z}$: general arccotangent function and $\operatorname{Arctan}\NVar{z}$: general arctangent function A&S Ref: 4.4.36 Permalink: http://dlmf.nist.gov/4.24.E17 Encodings: TeX, pMML, png See also: Annotations for §4.24(iii), §4.24 and Ch.4

The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. All square roots have either possible value.