Inverse function identities · proof ·what it is and when to use it
Proof: sin(arcsin x) = x
Sine of an inverse sine — proved by unfold the definition. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Set the inner inverse equal to an angle and translate the definition into a sine statement. Everything then follows from that single sentence.
- Let θ = arcsin xName the angle the inverse function returns.
- sin θ = x, with θ in [−π/2, π/2]That is what arcsin means.
- sin(arcsin x) = sin θ = xSubstitute θ back.
The restriction −1 ≤ x ≤ 1 is not decoration: outside it there is no angle whose sine is x, so arcsin x itself has no value.
Where the proof stops applying
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.
Outside that interval the inverse function returns nothing at all, so the statement is not false there — it has no meaning. The proof above never leaves the interval.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| 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 |
Related
- How would I find sin(arcsin x) = x myself? — the derivation, which is a different question from the proof.
- Sine of an inverse sine: when to use it — the practical side.
- All inverse function identities · proof index