Inverse function identities · trigonometric identity
sin(arcsin x) = x — Sine of an inverse sine
When to use it
Whenever an inverse function is wrapped in its own function: arcsin x already IS an angle, and taking its sine just returns the number you put in. The reverse order — arcsin(sin x) — is NOT this identity.
Why it is true
arcsin x is defined as the angle in [−π/2, π/2] whose sine is x. Applying sine to that angle gives back x by definition — nothing is proved here, the notation is doing the work.
The full line-by-line version is on the proof page for sine of an inverse sine; the “how would I find this myself” version is in the derivation.
Where it comes from
Solve sin θ = x for θ, then substitute. Start from the equation an inverse function was invented to answer, rather than from the definition of the inverse.
- sin θ = xThe equation whose solution is wanted.
- θ = arcsin xName the solution in [−π/2, π/2].
- sin(arcsin x) = xPut the second line into the first.
Same result, different direction of travel: this route explains why the identity cannot have a sign choice.
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 | -1 | yes |
| -0.8 | -0.8 | -0.8 | yes |
| -0.6 | -0.6 | -0.6 | yes |
| -0.4 | -0.4 | -0.4 | yes |
| -0.2 | -0.2 | -0.2 | yes |
| 0 | 0 | 0 | yes |
| 0.2 | 0.2 | 0.2 | yes |
| 0.4 | 0.4 | 0.4 | yes |
| 0.6 | 0.6 | 0.6 | yes |
| 0.8 | 0.8 | 0.8 | yes |
| 1 | 1 | 1 | 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
Reading it as arcsin(sin x) = x. That one is false outside [−π/2, π/2]: arcsin(sin 120°) = −60°, not 120°, because arcsin only answers with an angle in its own range.
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
- cos(arccos x) = xCosine of an inverse cosinedetailsproof
- arcsin x + arccos x = π/2Inverse sine plus inverse cosinedetailsproof
- cos(arcsin x) = √(1 − x²)Cosine of an inverse sinedetailsproof
Category hub: Inverse function identities · all identities: /identities