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4 Elementary FunctionsTrigonometric Functions

§4.24 Inverse Trigonometric Functions: Further Properties

Contents
  1. §4.24(i) Power Series
  2. §4.24(ii) Derivatives
  3. §4.24(iii) Addition Formulas

§4.24(i) Power Series

4.24.1 arcsinz=z+12z33+1324z55+135246z77+,
|z|1.
4.24.2 arccosz=(2(1z))1/2(1+n=1135(2n1)22n(2n+1)n!(1z)n),
|1z|2.
4.24.3 arctanz=zz33+z55z77+,
|z|1, z±i.
4.24.4 arctanz=±π21z+13z315z5+,
z0, |z|1.
4.24.5 arctanz=zz2+1(1+23z21+z2+2435(z21+z2)2+),
(z2)>12,

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

4.24.6 x2y2=12.

§4.24(ii) Derivatives

4.24.7 ddzarcsinz =(1z2)1/2,
4.24.8 ddzarccosz =(1z2)1/2,
4.24.9 ddzarctanz =11+z2.
4.24.10 ddzarccscz =1z(z21)1/2,
z0.
4.24.11 ddzarcsecz =±1z(z21)1/2,
z0.
4.24.12 ddzarccotz =11+z2.

§4.24(iii) Addition Formulas

4.24.13 Arcsinu±Arcsinv=Arcsin(u(1v2)1/2±v(1u2)1/2),
4.24.14 Arccosu±Arccosv=Arccos(uv((1u2)(1v2))1/2),
4.24.15 Arctanu±Arctanv=Arctan(u±v1uv),
4.24.16 Arcsinu±Arccosv=Arcsin(uv±((1u2)(1v2))1/2)=Arccos(v(1u2)1/2u(1v2)1/2),
4.24.17 Arctanu±Arccotv=Arctan(uv±1vu)=Arccot(vuuv±1).

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.