Definitions, domains and ranges
Sine, cosine and tangent repeat, so they are only invertible on an interval where each value occurs once. y = arcsin x means sin y = x with y in [−π/2, π/2]; y = arccos x means cos y = x with y in [0, π]; y = arctan x means tan y = x with y in (−π/2, π/2). They are also written sin⁻¹ x, cos⁻¹ x and tan⁻¹ x, where ⁻¹ means inverse function, not 1/sin x.
| Function | Domain | Range (principal values) | Graph |
|---|---|---|---|
| arcsin x | [−1, 1] | [−π/2, π/2] | Increasing S-shaped curve from (−1, −π/2) through (0, 0) to (1, π/2); vertical tangents at the ends |
| arccos x | [−1, 1] | [0, π] | Decreasing curve from (−1, π) through (0, π/2) to (1, 0); the arcsin graph reflected and shifted, arccos x = π/2 − arcsin x |
| arctan x | All real numbers | (−π/2, π/2) | Increasing curve through (0, 0) with horizontal asymptotes y = −π/2 and y = π/2 |
Sources: OpenStax: Precalculus 2e §6.3 Inverse Trigonometric Functions; NIST Digital Library of Mathematical Functions: DLMF §4.23 Inverse Trigonometric Functions
Derivatives and integrals
Read in reverse, the derivatives give the standard integrals that produce inverse trig functions: ∫ dx ÷ √(a² − x²) = arcsin(x/a) + C and ∫ dx ÷ (a² + x²) = (1/a)·arctan(x/a) + C for a > 0.
| Function | Derivative | Integral |
|---|---|---|
| arcsin x | 1 ÷ √(1 − x²), |x| < 1 | x·arcsin x + √(1 − x²) + C |
| arccos x | −1 ÷ √(1 − x²), |x| < 1 | x·arccos x − √(1 − x²) + C |
| arctan x | 1 ÷ (1 + x²) | x·arctan x − ½ ln(1 + x²) + C |
Sources: OpenStax: Calculus Volume 1 §3.7 Derivatives of Inverse Functions; OpenStax: Calculus Volume 1 §5.7 Integrals Resulting in Inverse Trigonometric Functions; NIST Digital Library of Mathematical Functions: DLMF §4.26 Trigonometric Functions: Integrals
Compositions
Each composition of a trig function with an inverse trig function simplifies to an algebraic expression. Draw a right triangle whose angle θ = arcsin x, arccos x or arctan x, label two sides from x, and find the third side with the Pythagorean theorem.
| Expression | Simplifies to | Valid for |
|---|---|---|
| sin(arcsin x) | x | −1 ≤ x ≤ 1 |
| cos(arcsin x) | √(1 − x²) | −1 ≤ x ≤ 1 |
| tan(arcsin x) | x ÷ √(1 − x²) | −1 < x < 1 |
| sin(arccos x) | √(1 − x²) | −1 ≤ x ≤ 1 |
| cos(arccos x) | x | −1 ≤ x ≤ 1 |
| tan(arccos x) | √(1 − x²) ÷ x | −1 ≤ x ≤ 1, x ≠ 0 |
| sin(arctan x) | x ÷ √(1 + x²) | All real x |
| cos(arctan x) | 1 ÷ √(1 + x²) | All real x |
| tan(arctan x) | x | All real x |
| arcsin(sin x) | x | Only for −π/2 ≤ x ≤ π/2 |
| arccos(cos x) | x | Only for 0 ≤ x ≤ π; in general |x − 2πk|, with k the integer nearest x ÷ 2π |
| arctan(tan x) | x | Only for −π/2 < x < π/2 |
| arccos(sin x) | π/2 − x | For −π/2 ≤ x ≤ π/2 (since arccos y = π/2 − arcsin y) |
| arcsin(cos x) | π/2 − x | For 0 ≤ x ≤ π |
Sources: OpenStax: Precalculus 2e §6.3 Inverse Trigonometric Functions; NIST Digital Library of Mathematical Functions: DLMF §4.24 Inverse Trigonometric Functions: Further Properties