Inverse function identities · trigonometric identity
sin(arccos x) = √(1 − x²) — Sine of an inverse cosine
When to use it
Same shape as cos(arcsin x) and the reason both are written without a ±: the angle arccos returns lives in [0, π], where sine is never negative either.
Why it is true
With θ = arccos x we have cos θ = x and θ ∈ [0, π]. Then sin²θ = 1 − x² and sine is ≥ 0 on [0, π], so the positive root is forced.
The full line-by-line version is on the proof page for sine of an inverse cosine; the “how would I find this myself” version is in the derivation.
Where it comes from
Transfer it through the complementary-angle identity. If you already trust arcsin x + arccos x = π/2, this falls out without any new algebra.
- arccos x = π/2 − arcsin xComplementary-angle identity. · Inverse sine plus inverse cosine
- sin(arccos x) = sin(π/2 − arcsin x)Substitute.
- = cos(arcsin x) = √(1 − x²)Cofunction, then the known cosine form. · Cofunction identity for sine, Cosine of an inverse sine
This is the cheapest way to remember the pair: one of them, plus the complementary-angle identity, gives the other.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Value of x | Left side | Right side | Agree |
|---|---|---|---|
| -1 | 0 | 0 | yes |
| -0.8 | 0.6 | 0.6 | yes |
| -0.6 | 0.8 | 0.8 | yes |
| -0.4 | 0.916515 | 0.916515 | yes |
| -0.2 | 0.979796 | 0.979796 | yes |
| 0 | 1 | 1 | yes |
| 0.2 | 0.979796 | 0.979796 | yes |
| 0.4 | 0.916515 | 0.916515 | yes |
| 0.6 | 0.8 | 0.8 | yes |
| 0.8 | 0.6 | 0.6 | yes |
| 1 | 0 | 0 | yes |
What x is on this page
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.
This is the one family on the site where the letter is not an angle. Every other page samples angles across [−4π, 4π]; here the sampler stays inside the interval above, because outside it the inverse function has no value to compare.
The mistake students make
Expecting a different formula from cos(arcsin x) and being confused when it looks identical. The two compositions are different angles, but both land on the same positive root — the expressions are equal, the reasoning is not.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for inverse function identities.
Related identities
- cos(arcsin x) = √(1 − x²)Cosine of an inverse sinedetailsproof
- arcsin x + arccos x = π/2Inverse sine plus inverse cosinedetailsproof
- cos(arccos x) = xCosine of an inverse cosinedetailsproof
Category hub: Inverse function identities · all identities: /identities