Inverse function identities · derivation ·proof
Where arccos(−x) = π − arccos x comes from
Transfer it from arcsin through the complementary identity. If arcsin x + arccos x = π/2 is already trusted, the negative-input rule for arccos needs two lines and no new facts.
Building it step by step
- arccos(−x) = π/2 − arcsin(−x)Complementary identity. · Inverse sine plus inverse cosine
- = π/2 + arcsin xarcsin is odd. · Inverse sine of a negative
- = π − arccos xBecause arcsin x = π/2 − arccos x.
Two odd-looking rules in the family, one fact behind them: the two inverse functions are complementary.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
- arcsin(−x) = −arcsin x — Inverse sine of a negative
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
Where this route is used
When a negative cosine has to be expressed through the positive case — for example turning cos θ = −0.4 into a reference angle plus π.
- simplify
- solve
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 | 0 | 0 | yes |
| -0.8 | 0.643501 | 0.643501 | yes |
| -0.6 | 0.927295 | 0.927295 | yes |
| -0.4 | 1.159279 | 1.159279 | yes |
| -0.2 | 1.369438 | 1.369438 | yes |
| 0 | 1.570796 | 1.570796 | yes |
Neighbouring derivations
- arcsin(−x) = −arcsin x — Inverse sine of a negative
- arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
- cos(−θ) = cos θ — Cosine is even
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from arcsin x + arccos x = π/2 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the inverse cosine of a negative page.