Identity library · 8 formulas
Inverse function identities
Inverse functions are angles, not numbers — but their INPUT is a number between −1 and 1. That single fact decides everything in this family: which compositions collapse to x, why cos(arcsin x) is the positive root, and why arcsin(−x) = −arcsin x while arccos(−x) = π − arccos x. Note the order: sin(arcsin x) = x always, arcsin(sin x) = x only on [−π/2, π/2].
Every inverse function identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin(arcsin x) = x | Sine of an inverse sine | simplify, verify |
| cos(arccos x) = x | Cosine of an inverse cosine | simplify, verify |
| cos(arcsin x) = √(1 − x²) | Cosine of an inverse sine | simplify, integrate |
| sin(arccos x) = √(1 − x²) | Sine of an inverse cosine | simplify, integrate |
| tan(arcsin x) = x / √(1 − x²) | Tangent of an inverse sine | simplify, integrate |
| arcsin x + arccos x = π/2 | Inverse sine plus inverse cosine | simplify, verify, solve |
| arcsin(−x) = −arcsin x | Inverse sine of a negative | simplify, verify |
| arccos(−x) = π − arccos x | Inverse cosine of a negative | simplify, solve |
Inverse function identities (8)
arcsin and arccos: what survives when a function wraps its own inverse.
| Identity | Name | proof |
|---|---|---|
| sin(arcsin x) = x | Sine of an inverse sine | proof |
| cos(arccos x) = x | Cosine of an inverse cosine | proof |
| cos(arcsin x) = √(1 − x²) | Cosine of an inverse sine | proof |
| sin(arccos x) = √(1 − x²) | Sine of an inverse cosine | proof |
| tan(arcsin x) = x / √(1 − x²) | Tangent of an inverse sine | proof |
| arcsin x + arccos x = π/2 | Inverse sine plus inverse cosine | proof |
| arcsin(−x) = −arcsin x | Inverse sine of a negative | proof |
| arccos(−x) = π − arccos x | Inverse cosine of a negative | proof |
Doing something with them
- Which identity should I use? — describe an expression, get the rule.
- Verify — check a claim, see a counterexample if it is false.
- Printable cheat sheet — this family pre-selected.