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4 Elementary FunctionsTrigonometric Functions

§4.24 Inverse Trigonometric Functions: Further Properties


§4.24(i) Power Series

4.24.1 arcsinz=z+12z33+1324z55+135246z77+,
4.24.2 arccosz=(2(1-z))1/2×(1+n=1135(2n-1)22n(2n+1)n!(1-z)n),
4.24.3 arctanz=z-z33+z55-z77+,
|z|1, z±i.
4.24.4 arctanz=±π2-1z+13z3-15z5+,
z0, |z|1.
4.24.5 arctanz=zz2+1(1+23z21+z2+2435(z21+z2)2+),

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

4.24.6 x2-y2=-12.

§4.24(ii) Derivatives

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

§4.24(iii) Addition Formulas

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