Pascal, ≥ 162 bytes

“Extended Pascal” (ISO standard 10206) defines that complex numbers are part of the language.

function s(x:complex):complex;begin s:=cmplx(0,−1)*ln(sqrt(1−x*x)+cmplx(0,1)*x)end

    function arcsin(protected x: complex): complex;
        begin
            arcsin := cmplx(0, −1) * ln(sqrt(1 − sqr(x)) + cmplx(0, 1) * x);
        end;

 function c(x:complex):complex;begin c:=cmplx(0,−1)*ln(sqrt(x*x−1)+x)end

     function arccos(protected x: complex): complex;
        begin
            arccos := cmplx(0, −1) * ln(sqrt(sqr(x) − 1) + x);
        end;

Arc tangent is already part of the language (built-in function arctan).

arctan(x)

Pascal, 162 194 bytes

“Extended Pascal” (ISO standard 10206) defines that complex numbers are part of the language. When calling one of these functions, a real argument is automatically promoted to a complex value (because the formal parameter list says the function requires a complex number).

const i=cmplx(0,1);type r=complex;function s(x:r):r;begin s:=−i*ln(sqrt(1−x*x)+i*x);end;function c(x:r):r;begin c:=−i*ln(sqrt(x*x−1)+x);end;function t(x:r):r;begin t:=i/2*ln((1−i*x)/(1+i*x));end

Ungolfed:

    const
        i = cmplx(0, 1);
    
    function arcsin(protected x: complex): complex;
        begin
            arcsin := −i * ln(sqrt(1 − sqr(x)) + i * x);
        end;
    
    function arccos(protected x: complex): complex;
        begin
            arccos := −i * ln(sqrt(sqr(x) − 1) + x);
        end;
    
    function arctan(protected x: complex): complex;
        begin
            arctan := i / 2 * ln((1 − i * x) / (1 + i * x));
        end;

Arc tangent is actually already part of the language. Both in ISO standard 7185 (“Standard Pascal”) and ISO standard 10206 (“Extended Pascal”) the built-in function arctan is available. Since complex numbers are defined by EP, only the latter accepts and can return complex values.