Inverse function identities · proof ·what it is and when to use it
Proof: arcsin x + arccos x = π/2
Inverse sine plus inverse cosine — proved by take the sine of both sides. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Show the two sides have the same sine and lie in the same interval, so they must be equal.
- Let α = arcsin x, β = arccos xSo sin α = x and cos β = x.
- sin α = cos β = sin(π/2 − β)Cofunction identity. · Cofunction identity for sine
- α = π/2 − βBoth α and π/2 − β sit in [−π/2, π/2], where sine is one-to-one.
- arcsin x + arccos x = π/2Rearrange.
The one-to-one step is doing real work: without the range restriction, equal sines would only give equal-or-supplementary angles.
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 | 1.570796 | 1.570796 | yes |
| -0.8 | 1.570796 | 1.570796 | yes |
| -0.6 | 1.570796 | 1.570796 | yes |
| -0.4 | 1.570796 | 1.570796 | yes |
| -0.2 | 1.570796 | 1.570796 | yes |
| 0 | 1.570796 | 1.570796 | yes |
Related
- How would I find arcsin x + arccos x = π/2 myself? — the derivation, which is a different question from the proof.
- Inverse sine plus inverse cosine: when to use it — the practical side.
- All inverse function identities · proof index