Inverse function identities · derivation ·proof
Where sin(arcsin x) = x 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.
Building it step by step
- 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.
Where this route is used
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.
- 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 | -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 |
Neighbouring derivations
- cos(arccos x) = x — Cosine of an inverse cosine
- arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
- cos(arcsin x) = √(1 − x²) — Cosine of an inverse sine
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(arcsin(x)) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sine of an inverse sine page.