Inverse Trigonometric Functions Calculator

Enter a value and choose an inverse trig function (arcsin, arccos, arctan, arccsc, arcsec, or arccot) to get the principal angle in degrees and radians.

Quick Facts

arcsin & arccos
Domain [-1, 1]
Ranges are [-90°, 90°] for arcsin and [0°, 180°] for arccos.
arctan
Domain: all real numbers
Range is (-90°, 90°), never reaching the endpoints.
arccsc & arcsec
Domain |x| ≥ 1
Defined as arcsin(1/x) and arccos(1/x).
Identity
arcsin(x) + arccos(x) = 90°
Holds for every x in [-1, 1].

Your Results

Calculated
Angle (degrees)
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Principal value in degrees
Angle (radians)
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Principal value in radians
Expression
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Function and input evaluated
Verification
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Forward function applied to the angle

Ready

Enter a value and choose a function, then press Calculate.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions ("arc" functions) undo what sine, cosine, and tangent do: instead of taking an angle and returning a ratio, they take a ratio and return the angle that produces it. Because sin, cos, and tan are periodic — many angles share the same ratio — each inverse function is defined to return a single principal value from a restricted range, so the calculator always gives one well-defined answer rather than infinitely many.

Domains, ranges, and formulas

  • arcsin(x) = sin⁻¹(x) — domain [-1, 1], principal range [-90°, 90°]
  • arccos(x) = cos⁻¹(x) — domain [-1, 1], principal range [0°, 180°]
  • arctan(x) = tan⁻¹(x) — domain: all real numbers, principal range (-90°, 90°)
  • arccsc(x) = arcsin(1/x) — domain |x| ≥ 1, principal range [-90°, 90°] excluding 0°
  • arcsec(x) = arccos(1/x) — domain |x| ≥ 1, principal range [0°, 180°] excluding 90°
  • arccot(x) = 90° − arctan(x) — domain: all real numbers, principal range (0°, 180°)

This calculator evaluates the chosen function with JavaScript's built-in Math.asin, Math.acos, and Math.atan (which return radians), converts the result to degrees by multiplying by 180/π, and enforces each function's domain before computing — so an out-of-range input (like arcsin(2)) is rejected instead of silently returning a wrong number.

Degrees vs. radians

Both units describe the same angle: radians = degrees × π/180, and a full circle is 2π radians = 360°. This calculator reports both so you can plug the result directly into a geometry problem (degrees) or a calculus/physics formula (radians, since derivatives and Taylor series for trig functions assume radian input).

Frequently Asked Questions

What are the domain and range of arcsin, arccos, and arctan?
arcsin(x) and arccos(x) accept x from -1 to 1. arcsin returns a principal angle from -90° to 90°, while arccos returns a principal angle from 0° to 180°. arctan(x) accepts any real number and returns a principal angle from -90° to 90° (exclusive).
Why do arcsin and arccos require inputs between -1 and 1?
Sine and cosine only ever output values between -1 and 1, so their inverses can only accept inputs in that same range. Entering a value like 2 has no real angle whose sine or cosine equals 2, so the calculator flags it as invalid.
How are arccsc(x) and arcsec(x) calculated?
arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x), both defined only when |x| ≥ 1 (since cosecant and secant never fall strictly between -1 and 1). arccsc returns an angle in [-90°, 90°] excluding 0°, and arcsec returns an angle in [0°, 180°] excluding 90°.
How is arccot(x) defined, and why does its range differ from arctan?
This calculator uses the standard calculus convention arccot(x) = 90° − arctan(x), which is continuous for every real x and has range (0°, 180°). This differs from simply taking arctan(1/x), which is undefined at x = 0 and has a jump discontinuity there.

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