Inverse function identities · proof ·what it is and when to use it
Proof: cos(arccos x) = x
Cosine of an inverse cosine — proved by definition, with the range checked. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Identical in shape to the sine version; the only thing worth watching is which interval arccos is allowed to answer with.
- Let θ = arccos xName the returned angle.
- cos θ = x, with θ in [0, π]Definition of arccos.
- cos(arccos x) = xSubstitute.
The range [0, π] is exactly where cosine takes each value once — that is what makes arccos a function at all.
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 cos(arccos x) = x myself? — the derivation, which is a different question from the proof.
- Cosine of an inverse cosine: when to use it — the practical side.
- All inverse function identities · proof index