Inverse function identities · trigonometric identity
cos(arcsin x) = √(1 − x²) — Cosine of an inverse sine
When to use it
The workhorse of calculus substitution: an angle is known only through its sine and you need its cosine. It also appears whenever a right triangle is drawn from sin θ = x.
Why it is true
sin²θ + cos²θ = 1 with θ = arcsin x gives cos²θ = 1 − x². The square root is taken positive because arcsin x always lies in [−π/2, π/2], where cosine is never negative.
The full line-by-line version is on the proof page for cosine of an inverse sine; the “how would I find this myself” version is in the derivation.
Where it comes from
Draw the right triangle. The fastest route in an exam: build a triangle from the given ratio and read the missing side off Pythagoras.
- Opposite = x, hypotenuse = 1Because sin θ = x/1.
- Adjacent = √(1 − x²)Pythagoras on the triangle.
- cos θ = adjacent / hypotenuse = √(1 − x²)Definition of cosine.
The triangle silently assumes x ≥ 0; the range argument in the proof is what makes the formula valid for negative x too.
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
Dropping the range argument and writing ±√(1 − x²). For this composition the sign is fixed: cos(arcsin 0.5) = 0.866…, never −0.866…. The same carelessness in the opposite direction (arcsin of a cosine) is where students lose marks.
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
- sin(arcsin x) = xSine of an inverse sinedetailsproof
- tan(arcsin x) = x / √(1 − x²)Tangent of an inverse sinedetailsproof
- sin(arccos x) = √(1 − x²)Sine of an inverse cosinedetailsproof
Category hub: Inverse function identities · all identities: /identities