Definition and principal branch. A trigonometric function is not one-one on its full domain (periodicity), so its inverse is defined only after restricting to a branch on which it is bijective; the conventional choice is the principal value branch, and sin−1x etc. always means the value on this branch, never (sinx)−1.
Principal value branches.
sin−1x:[−2π,2π],cos−1x:[0,π],tan−1x:(−2π,2π),
cot−1x:(0,π),sec−1x:[0,π]−{2π},cosec−1x:[−2π,2π]−{0}.
Domain and range. sin−1x,cos−1x: domain [−1,1] (sine and cosine never exceed 1 in magnitude). tan−1x,cot−1x: domain R (tangent and cotangent take every real value). sec−1x,cosec−1x: domain R−(−1,1) (secant and cosecant never lie strictly between −1 and 1, from sec2θ≥1).
Graphs. Each inverse trigonometric function's graph is the reflection, in the line y=x, of the corresponding trigonometric function's graph restricted to its principal branch. sin−1x and tan−1x are strictly increasing and odd; cos−1x is strictly decreasing; tan−1x has horizontal asymptotes y=±2π.
Elementary properties.
sin−1(x1)=cosec−1x,cos−1(x1)=sec−1x,tan−1(x1)=cot−1x (x>0);
sin−1(−x)=−sin−1x,tan−1(−x)=−tan−1x,cosec−1(−x)=−cosec−1x;
cos−1(−x)=π−cos−1x,sec−1(−x)=π−sec−1x,cot−1(−x)=π−cot−1x;
sin−1x+cos−1x=2π,tan−1x+cot−1x=2π,sec−1x+cosec−1x=2π; …