Inverse function identities · trigonometric identity
arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
When to use it
Whenever both inverse functions appear in one expression — usually to swap one for the other, to simplify a sum of angles, or to check that two answers in a textbook are the same number.
Why it is true
The two acute angles of a right triangle add to π/2, and arcsin x / arccos x are precisely those two angles for a triangle whose side ratio is x.
The full line-by-line version is on the proof page for inverse sine plus inverse cosine; the “how would I find this myself” version is in the derivation.
Where it comes from
Two acute angles of one right triangle. Draw the triangle once and the identity is visible before it is written down.
- Right triangle, opposite side x, hypotenuse 1Sets sin of one acute angle to x.
- That acute angle = arcsin xDefinition.
- The other acute angle = arccos xIts cosine is the same ratio x.
- arcsin x + arccos x = π/2The two acute angles of a right triangle sum to 90°.
This is the version to keep in your head; the algebraic proof above is what extends it to negative x.
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 | 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 |
| 0.2 | 1.570796 | 1.570796 | yes |
| 0.4 | 1.570796 | 1.570796 | yes |
| 0.6 | 1.570796 | 1.570796 | yes |
| 0.8 | 1.570796 | 1.570796 | yes |
| 1 | 1.570796 | 1.570796 | 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
Treating the sum as 90° only for positive x. It holds across the whole domain: at x = −0.5, arcsin gives −30° and arccos gives 120° — still 90°.
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
- sin(arccos x) = √(1 − x²)Sine of an inverse cosinedetailsproof
- sin θ = cos(90° − θ)Cofunction identity for sinedetailsproof
Category hub: Inverse function identities · all identities: /identities