Inverse Trigonometric Calculator

Calculate all 6 inverse trig functions with interactive plots

Arcsine (arcsin)
30.0000°
sin⁻¹(x), x ∈ [-1, 1]
Arccosine (arccos)
60.0000°
cos⁻¹(x), x ∈ [-1, 1]
Arctangent (arctan)
26.5651°
tan⁻¹(x), x ∈ ℝ
Arccotangent (arccot)
63.4349°
cot⁻¹(x), x ∈ ℝ
Arcsecant (arcsec)
60.0000°
sec⁻¹(x), |x| ≥ 1
Arccosecant (arccsc)
30.0000°
csc⁻¹(x), |x| ≥ 1
Understanding Inverse Functions

Inverse trigonometric functions are used to find angles when the value of a trigonometric function is known.

  • arcsin(x) - angle whose sine is x (x ∈ [-1, 1])
  • arccos(x) - angle whose cosine is x (x ∈ [-1, 1])
  • arctan(x) - angle whose tangent is x (x ∈ ℝ)
  • arccot(x) - angle whose cotangent is x (x ∈ ℝ)
  • arcsec(x) - angle whose secant is x (|x| ≥ 1)
  • arccsc(x) - angle whose cosecant is x (|x| ≥ 1)
  • Range: arcsin: [-90°, 90°], arccos: [0°, 180°], arctan: (-90°, 90°)
  • Used in geometry, physics, and engineering
Inverse Trig Formulas
arcsin(x) = sin⁻¹(x)
arccos(x) = cos⁻¹(x)
arctan(x) = tan⁻¹(x)
arccot(x) = cot⁻¹(x)
arcsec(x) = sec⁻¹(x)
arccsc(x) = csc⁻¹(x)
sin(arcsin(x)) = x
arcsin(x) + arccos(x) = π/2
Inverse Trig Facts
  • Inverse trig functions are also called "arc functions"
  • They are the inverses of the basic trig functions
  • Results are typically given in radians in higher mathematics
  • Inverse trig functions are used in calculus for integration
  • They have applications in navigation and robotics

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the trigonometric functions.
arcsin(x)
Also written as sin⁻¹(x), it gives the angle whose sine is x.
Domain: [-1, 1]
Range: [-90°, 90°] or [-π/2, π/2] radians
arccos(x)
Also written as cos⁻¹(x), it gives the angle whose cosine is x.
Domain: [-1, 1]
Range: [0°, 180°] or [0, π] radians
arctan(x)
Also written as tan⁻¹(x), it gives the angle whose tangent is x.
Domain: (-∞, ∞)
Range: (-90°, 90°) or (-π/2, π/2) radians
arccsc(x)
Also written as csc⁻¹(x), it gives the angle whose cosecant is x.
Domain: (-∞, -1] ∪ [1, ∞)
Range: [-90°, 0°) ∪ (0°, 90°] or [-π/2, 0) ∪ (0, π/2] radians
arcsec(x)
Also written as sec⁻¹(x), it gives the angle whose secant is x.
Domain: (-∞, -1] ∪ [1, ∞)
Range: [0°, 90°) ∪ (90°, 180°] or [0, π/2) ∪ (π/2, π] radians
arccot(x)
Also written as cot⁻¹(x), it gives the angle whose cotangent is x.
Domain: (-∞, ∞)
Range: (0°, 180°) or (0, π) radians

Properties and Identities

Key Properties:

  • arcsin(x) + arccos(x) = π/2 (90°)
  • arctan(x) + arctan(1/x) = π/2 for x > 0
  • arctan(x) + arctan(1/x) = -π/2 for x < 0
  • arcsin(x) = arccos(√(1-x²)) for x in [0,1]
  • arccos(x) = arcsin(√(1-x²)) for x in [0,1]

Derivatives:

  • 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)

Integrals:

  • ∫ arcsin(x) dx = x arcsin(x) + √(1-x²) + C
  • ∫ arccos(x) dx = x arccos(x) - √(1-x²) + C
  • ∫ arctan(x) dx = x arctan(x) - ½ ln(1+x²) + C
Common Values
Function Value Angle (deg) Angle (rad)
arcsin(0) 0 0° 0
arcsin(0.5) 0.5 30° π/6
arcsin(√2/2) 0.7071 45° π/4
arcsin(1) 1 90° π/2
arccos(0) 0 90° π/2
arccos(0.5) 0.5 60° π/3
arccos(√2/2) 0.7071 45° π/4
arccos(1) 1 0° 0
arctan(0) 0 0° 0
arctan(1/√3) 0.5774 30° π/6
arctan(1) 1 45° π/4
arctan(√3) 1.7321 60° π/3

Frequently Asked Questions

Inverse trigonometric functions are the inverse functions of the basic trigonometric functions. They are used to find angles when given trigonometric ratios.

For example:

  • arcsin(0.5) = 30° or π/6 radians
  • arccos(0.5) = 60° or π/3 radians
  • arctan(1) = 45° or π/4 radians

These functions are also known as arc functions or anti-trigonometric functions.

Inverse trigonometric functions have restricted ranges because trigonometric functions are periodic and not one-to-one over their entire domains.

To make them invertible, we restrict their domains:

  • For arcsin, we use [-π/2, π/2]
  • For arccos, we use [0, π]
  • For arctan, we use (-π/2, π/2)

These restrictions ensure that each input value corresponds to exactly one output value, making the functions well-defined and invertible.

To calculate inverse trigonometric functions:

  1. Identify which function you need (arcsin, arccos, arctan, etc.)
  2. Ensure your input value is within the domain of the function
  3. Use a calculator or mathematical software
  4. Determine if you need the result in degrees or radians

For example, to calculate arcsin(0.5):

  • arcsin(0.5) = 30° or π/6 radians
  • This is because sin(30°) = 0.5

For values not on the unit circle, you'll typically need a calculator.

There is no difference between arcsin and sin⁻¹ - they are two notations for the same function.

Both represent the inverse sine function:

  • arcsin(x) is the angle whose sine is x
  • sin⁻¹(x) is the inverse function of sin(x)

The same applies to other inverse trig functions:

  • arccos(x) = cos⁻¹(x)
  • arctan(x) = tan⁻¹(x)
  • arccsc(x) = csc⁻¹(x)
  • arcsec(x) = sec⁻¹(x)
  • arccot(x) = cot⁻¹(x)

The "arc" prefix comes from the fact that these functions give the arc length on the unit circle corresponding to a given ratio.

Inverse trigonometric functions are used whenever you need to find an angle from a trigonometric ratio:

  • Geometry: Finding angles in triangles when sides are known
  • Physics: Calculating angles in vector problems
  • Engineering: Determining phase angles in AC circuits
  • Navigation: Calculating bearings and directions
  • Computer Graphics: Determining rotation angles
  • Calculus: Solving integrals and differential equations
  • Signal Processing: Analyzing phase shifts

Any situation where you have a trigonometric ratio and need to find the corresponding angle requires inverse trig functions.