Inverse function identities · trigonometric identity
arcsin(−x) = −arcsin x — Inverse sine of a negative
When to use it
To pull a minus sign out of an inverse sine — common when integrating or when comparing two answers that differ only by a sign.
Why it is true
Sine is odd, and arcsin answers inside [−π/2, π/2], which is symmetric about 0. So the angle for −x is the negated angle for x.
The full line-by-line version is on the proof page for inverse sine of a negative; the “how would I find this myself” version is in the derivation.
Where it comes from
Reflect the graph. A picture route: oddness of sine turns into oddness of its inverse.
- 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.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Value of x | 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 |
| 0.2 | -0.201358 | -0.201358 | yes |
| 0.4 | -0.411517 | -0.411517 | yes |
| 0.6 | -0.643501 | -0.643501 | yes |
| 0.8 | -0.927295 | -0.927295 | yes |
| 1 | -1.570796 | -1.570796 | yes |
What x is on this page
x is a number from −1 to 1 — arcsin and arccos are only defined for inputs in this interval — outside it there is no angle to take.
This is the one family on the site where the letter is not an angle. Every other page samples angles across [−4π, 4π]; here the sampler stays inside the interval above, because outside it the inverse function has no value to compare.
The mistake students make
Applying the same rule to arccos. It is false there: arccos(−x) = π − arccos x, not −arccos x — arccos never answers with a negative angle.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for inverse function identities.
Related identities
- arccos(−x) = π − arccos xInverse cosine of a negativedetailsproof
- sin(arcsin x) = xSine of an inverse sinedetailsproof
- sin(−θ) = −sin θSine is odddetailsproof
Category hub: Inverse function identities · all identities: /identities