arcsin and arccos of common values
arcsin returns an angle in [−π/2, π/2] and arccos an angle in [0, π], so every argument in [−1, 1] has exactly one principal value. arcsin x + arccos x = π/2 for every x in [−1, 1].
| x | arcsin x | arcsin (rad) | arcsin (°) | arccos x | arccos (rad) | arccos (°) |
|---|---|---|---|---|---|---|
| −1 | −π/2 | −1.570796 | −90° | π | 3.141593 | 180° |
| −√3/2 | −π/3 | −1.047198 | −60° | 5π/6 | 2.617994 | 150° |
| −√2/2 | −π/4 | −0.785398 | −45° | 3π/4 | 2.356194 | 135° |
| −1/2 | −π/6 | −0.523599 | −30° | 2π/3 | 2.094395 | 120° |
| 0 | 0 | 0.000000 | 0° | π/2 | 1.570796 | 90° |
| 1/2 | π/6 | 0.523599 | 30° | π/3 | 1.047198 | 60° |
| √2/2 (= 1/√2) | π/4 | 0.785398 | 45° | π/4 | 0.785398 | 45° |
| √3/2 | π/3 | 1.047198 | 60° | π/6 | 0.523599 | 30° |
| 1 | π/2 | 1.570796 | 90° | 0 | 0.000000 | 0° |
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.23 Inverse Trigonometric Functions; OpenStax: Precalculus 2e §6.3 Inverse Trigonometric Functions
arctan of common values
arctan is defined for every real number and returns an angle strictly between −π/2 and π/2.
| x | arctan x | Radians | Degrees |
|---|---|---|---|
| −√3 | −π/3 | −1.047198 | −60° |
| −1 | −π/4 | −0.785398 | −45° |
| −√3/3 (= −1/√3) | −π/6 | −0.523599 | −30° |
| 0 | 0 | 0.000000 | 0° |
| √3/3 (= 1/√3) | π/6 | 0.523599 | 30° |
| 1 | π/4 | 0.785398 | 45° |
| √3 | π/3 | 1.047198 | 60° |
| 2 | No simple exact form | 1.107149 | 63.4349° |
| 3 | No simple exact form | 1.249046 | 71.5651° |
| x → +∞ | → π/2 (limit) | → 1.570796 | → 90° |
| x → −∞ | → −π/2 (limit) | → −1.570796 | → −90° |
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.23 Inverse Trigonometric Functions
Arguments outside the domain: arccos 2, arcsin 3, arcsin ∞
Sine and cosine of a real angle always lie between −1 and 1, so arcsin x and arccos x have no real value when |x| > 1. arccos 2, arccos 3, arcsin 2 and arcsin 3 are undefined over the real numbers, and a calculator in real mode reports an error.
They do have complex values. For x > 1, arccos x = ±i·ln(x + √(x² − 1)), so arccos 2 = ±i·ln(2 + √3) ≈ ±1.316958i; real x > 1 lies on the branch cut, so the sign depends on the convention the software uses. Treat such results as complex numbers, not angles.
arcsin ∞ and arccos ∞ are likewise undefined, because x never grows beyond 1 in the domain. Only arctan has a limit at infinity: arctan x → π/2 as x → ∞.
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.23 Inverse Trigonometric Functions; NIST Digital Library of Mathematical Functions: DLMF §4.24 Inverse Trigonometric Functions: Further Properties