Principal Value Domain (Principal Branch)
Take sinx=21. It has infinitely many solutions: x=6π,65π,613π,−67π,… — every angle whose sine is 21. So if we want an inverse that returns a single angle for sin−1(0.5), we must first agree on one angle to report. A function is allowed only one output per input, and sinx over all of R is many-to-one — it fails the horizontal line test and cannot be inverted as it stands.
The idea: restrict to one clean interval
For each trigonometric ratio we restrict the angle to a single standard interval on which the function is one-to-one while still covering its entire range exactly once. On that interval the inverse becomes well-defined and single-valued. That interval — the set of angles the inverse is allowed to return — is the principal value branch (also called the principal value domain).
The interval is chosen to be strictly monotonic, to hit every output once, and to sit as close to 0 as possible. For sine that is [−2π,2π], where sin increases from −1 to 1.
The principal value branch of an inverse trig function is the interval of angles it returns — the restricted interval on which the original ratio is one-to-one and onto its range.
| Inverse function | Domain (allowed inputs x) | Principal value branch (angles returned) |
|---|
| sin−1x | [−1,1] | [−2π,2π] |
| cos−1x | [−1,1] | [0,π] |
| tan−1x | R | (−2π,2π) |
| cot−1x | R | (0,π) |
| sec−1x | (−∞,−1]∪[1,∞) | [0,π]∖{2π} |
| csc−1x | (−∞,−1]∪[1,∞) | [−2π,2π]∖{0} |
Why the intervals differ …