Inverse function identities · derivation ·proof
Where arcsin(−x) = −arcsin x comes from
Reflect the graph. A picture route: oddness of sine turns into oddness of its inverse.
Building it step by step
- y = arcsin x is the reflection of y = sin x (x in [−π/2, π/2])How an inverse graph is made.
- That sine branch is symmetric about the originSine is odd on the whole line.
- Reflecting an origin-symmetric curve keeps it origin-symmetricSo arcsin is odd: arcsin(−x) = −arcsin x.
The same picture explains arccos: its branch is symmetric about the vertical line through the middle, not about the origin.
Where this route is used
To pull a minus sign out of an inverse sine — common when integrating or when comparing two answers that differ only by a sign.
- simplify
- verify
The result, checked at real values
Deriving a formula and testing it are different things; this table is the test. The numbers come from the same engine as the verifier, computed when the site was built.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| -1 | 1.570796 | 1.570796 | yes |
| -0.8 | 0.927295 | 0.927295 | yes |
| -0.6 | 0.643501 | 0.643501 | yes |
| -0.4 | 0.411517 | 0.411517 | yes |
| -0.2 | 0.201358 | 0.201358 | yes |
| 0 | 0 | 0 | yes |
Neighbouring derivations
- arccos(−x) = π − arccos x — Inverse cosine of a negative
- sin(arcsin x) = x — Sine of an inverse sine
- sin(−θ) = −sin θ — Sine is odd
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from arcsin(-x) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the inverse sine of a negative page.