Inverse function identities · trigonometric identity
arccos(−x) = π − arccos x — Inverse cosine of a negative
When to use it
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.
Why it is true
cos(π − θ) = −cos θ, and when θ runs over [0, π] so does π − θ. So the angle arccos assigns to −x is exactly π − arccos x.
The full line-by-line version is on the proof page for inverse cosine of a negative; the “how would I find this myself” version is in the derivation.
Where it 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.
- 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.
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 | 0.643501 | 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 |
| 0.2 | 1.772154 | 1.772154 | yes |
| 0.4 | 1.982313 | 1.982313 | yes |
| 0.6 | 2.214297 | 2.214297 | yes |
| 0.8 | 2.498092 | 2.498092 | yes |
| 1 | 3.141593 | 3.141593 | 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
Writing −arccos x, copying the arcsin rule. Test it at x = 1: arccos(−1) = π, while −arccos(1) = 0. The two are not close.
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
- arcsin(−x) = −arcsin xInverse sine of a negativedetailsproof
- arcsin x + arccos x = π/2Inverse sine plus inverse cosinedetailsproof
- cos(−θ) = cos θCosine is evendetailsproof
Category hub: Inverse function identities · all identities: /identities