The principal value of an inverse trigonometric function at a point x is the value of the inverse function at x that lies in its principal-value-branch range (the tables in the two concepts above). When solving (inverse function)(x)=y, there may be infinitely many angles θ with the right ratio, but exactly one of them lies in the principal range — that one is the principal value.
Tie-breaking rule. If two candidate values are numerically equal but opposite in sign (e.g. solving cosy=−21 might tempt y=±32π), the principal value is taken to be the positive one, subject to it actually lying in the function's principal range.
Reference table (principal domain → range of the inverse):
| Function | Principal domain | Range | Inverse | Domain | Range of principal value |
|---|
| sine | [−2π,2π] | [−1,1] | sin−1 | [−1,1] | [−2π,2π] |
| cosine | [0,π] | [−1,1] | cos−1 | [−1,1] | [0,π] |
| tangent | (−2π,2π) | R | tan−1 | R | (−2π,2π) |
| cosecant | [−2π,2π]∖{0} | R∖(−1,1) | cosec−1 | R∖(−1,1) | [−2π,2π]∖{0} |
| secant | [0,π]∖{2π} | R∖(−1,1) | sec−1 | R∖(−1,1) | [0,π]∖{2π} |
| cotangent | (0,π) | R | cot−1 | R | (0,π) |