Inverse Trigonometric Functions Calculator

Compute all six inverse trig functions at once: arcsin, arccos, arctan, arccsc, arcsec, and arccot.

Select Function & Enter Value

Enter a value to compute all applicable inverse trig functions. Functions with invalid inputs will show "undefined".

Result

Inverse Trig Functions for x = 0.5
6 functions
computed below
arcsin(0.5)
30°
π/6 rad
arccos(0.5)
60°
π/3 rad
arctan(0.5)
26.565°
0.4636 rad
arccsc(0.5)
undefined
|x| must be ≥ 1
arcsec(0.5)
undefined
|x| must be ≥ 1
arccot(0.5)
63.435°
1.1071 rad

Step-by-Step Solution

All inverse trig functions for x = 0.5

The Six Inverse Trigonometric Functions

Inverse trigonometric functions (also called arc-functions or anti-trigonometric functions) are the inverses of the standard trigonometric functions. They return the angle whose trig function value equals the input. Each has specific domain and range restrictions to ensure they are proper functions.

Summary of All Inverse Trig Functions

arcsin(x)

Domain: [-1, 1]. Range: [-π/2, π/2] or [-90°, 90°].

sin(θ) = x => θ = arcsin(x)

arccos(x)

Domain: [-1, 1]. Range: [0, π] or [0°, 180°].

cos(θ) = x => θ = arccos(x)

arctan(x)

Domain: all reals. Range: (-π/2, π/2) or (-90°, 90°).

tan(θ) = x => θ = arctan(x)

arccsc(x)

Domain: |x| ≥ 1. Range: [-π/2, π/2], x ≠ 0.

csc(θ) = x => θ = arccsc(x)

arcsec(x)

Domain: |x| ≥ 1. Range: [0, π], x ≠ π/2.

sec(θ) = x => θ = arcsec(x)

arccot(x)

Domain: all reals. Range: (0, π) or (0°, 180°).

cot(θ) = x => θ = arccot(x)

Important Identities

Complementary Relationships

  • arcsin(x) + arccos(x) = π/2 for all x in [-1, 1].
  • arctan(x) + arccot(x) = π/2 for all real x.
  • arcsec(x) + arccsc(x) = π/2 for all |x| ≥ 1.

Reciprocal Relationships

  • arccsc(x) = arcsin(1/x) for |x| ≥ 1.
  • arcsec(x) = arccos(1/x) for |x| ≥ 1.
  • arccot(x) = arctan(1/x) for x > 0.
  • arccot(x) = π + arctan(1/x) for x < 0.

Negative Argument Identities

  • arcsin(-x) = -arcsin(x) (odd function).
  • arccos(-x) = π - arccos(x).
  • arctan(-x) = -arctan(x) (odd function).
  • arccsc(-x) = -arccsc(x) (odd function).
  • arcsec(-x) = π - arcsec(x).
  • arccot(-x) = π - arccot(x).

Derivatives of Inverse Trig Functions

  • d/dx arcsin(x) = 1/√(1 - x²)
  • d/dx arccos(x) = -1/√(1 - x²)
  • d/dx arctan(x) = 1/(1 + x²)
  • d/dx arccsc(x) = -1/(|x|√(x² - 1))
  • d/dx arcsec(x) = 1/(|x|√(x² - 1))
  • d/dx arccot(x) = -1/(1 + x²)