Inverse Trigonometric Functions Calculator (Arcsin, Arccos, Arctan, Arcsec, Arccsc, Arccot

Inverse Trigonometric Functions, Principal Branches & Derivatives

Because standard circular trigonometric functions are periodic and non-monotonic, their domains must be restricted to define unique, single-valued inverse functions (principal branches):

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• Domains, Principal Ranges & Definitions**:

• (x) = ^-1(x): Domain [-1, 1], Principal Range [-(π) / (2), (π) / (2)] ([-90^, 90^])

• (x) = ^-1(x): Domain [-1, 1], Principal Range [0, π] ([0^, 180^])

• (x) = ^-1(x): Domain (-∞, ∞), Principal Range (-(π) / (2), (π) / (2)) ((-90^, 90^))

• arccsc(x) = : Domain |x| 1, Range [-(π) / (2), 0) = : Domain |x| 1, Range [0, (π) / (2)) = : Domain (-∞, ∞), Range (0, π)

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• Standard Derivatives of Inverse Trigonometric Functions**:

(d) / (dx) (x) = (1) / (√(1 - x^2)), (d) / (dx) (x) = -(1) / (√(1 - x^2))

(d) / (dx) (x) = (1) / (1 + x^2), (d) / (dx) arccot(x) = -(1) / (1 + x^2)

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• Standard Exact Trigonometric Angles**:

• (0) = 0^, = 30^, = 45^, (1) = 90^

• (0) = 90^, = 60^, (1) = 0^

• (0) = 0^, (1) = 45^, (√(3)) = 60^

How to Use This Calculator

Enter your Inverse Trigonometric Function — choose from arcsin(x) / sin⁻¹(x) [Domain: -1 ≤ x ≤ 1], arccos(x) / cos⁻¹(x) [Domain: -1 ≤ x ≤ 1], arctan(x) / tan⁻¹(x) [Domain: -∞ < x < ∞], Input Argument (x) (e.g. 0.5 in x), and Standard Mathematical Angle Benchmark — choose from Custom Argument (x), arcsin(0.5) = π/6 rad = 30.00°, arccos(√2/2 ≈ 0.7071) = π/4 rad = 45.00°. Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Inverse Trigonometric Functions Calculator (Arcsin, Arccos, Arctan, Arcsec, Arccsc, Arccot)

Input variables:

trigFunctionSelect — Inverse Trigonometric Function

inputArgumentX — Input Argument (x)

Computed outputs:

angleDegrees — Angle in Degrees (°)

angleRadians — Angle in Radians (rad)

angleGradians — Angle in Gradians (grad)

piMultipleRatio — Exact Multiple of π (θ/π)

firstDerivativeValue — First Derivative (d/dx f(x))

Mathematical relationships extracted from calculation logic:

deriv = Math.abs(x) === 1.0 ? Infinity : 1.0 / Math.sqrt(1.0 - x * x)

rad = Math.asin(1.0 / x)

deg = rad * (180.0 / Math.PI)

grad = rad * (200.0 / Math.PI)

piRatio = rad / Math.PI

Standard: NIST / ISO mathematical definitions

Worked Example

A civil surveyor calculating the slope angle of a ramp with a 1.0-meter rise over a 2.0-meter hypotenuse evaluates arcsin = arcsin(0.5) to determine the incline angle is exactly 30.0 degrees (0.5236 radians).

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.