MyTrigCalculator.com

Inverse Trig Calculator

Find the angle from a sine, cosine, or tangent value. Get results in degrees, radians, and π notation — with the valid principal range.

Enter Value

Tip

sin⁻¹(x) means "the angle whose sine is x" — not 1/sin(x). The -1 here is inverse-function notation, not an exponent.

Results

Pick a function and enter a value to find the angle

Inverse Trigonometric Functions Explained

The inverse trigonometric functions (also called arc functions or anti-trig functions) reverse the action of sine, cosine, and tangent. Where the regular trig functions take an angle and return a ratio, the inverse functions take a ratio and return an angle.

If sin(θ) = x, then arcsin(x) = θ

If cos(θ) = x, then arccos(x) = θ

If tan(θ) = x, then arctan(x) = θ

Notation

You'll see these written several ways: arcsin(x), asin(x), or sin⁻¹(x). They all mean the same inverse function. Important: sin⁻¹(x) does not mean 1/sin(x) — that would be cosecant. The "-1" is shorthand for "inverse function," not an exponent.

Domain restrictions

Sine and cosine only output values between -1 and 1, so their inverses only accept inputs in that range. Tangent's range is all real numbers, so arctan accepts any input.

  • arcsin: input ∈ [-1, 1], output ∈ [-90°, 90°] or [-π/2, π/2]
  • arccos: input ∈ [-1, 1], output ∈ [0°, 180°] or [0, π]
  • arctan: input ∈ ℝ (any real number), output ∈ (-90°, 90°) or (-π/2, π/2)

When to use each one

In a right triangle, you pick the inverse function based on which two sides you know:

  • Opposite + hypotenuse → arcsin(opp/hyp)
  • Adjacent + hypotenuse → arccos(adj/hyp)
  • Opposite + adjacent → arctan(opp/adj)

Worked example

A right triangle has opposite side 5 and adjacent side 12. The angle is:

tan(θ) = 5 / 12
θ = arctan(5/12) ≈ 22.62°

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Frequently Asked Questions

What is an inverse trig function?

Inverse trigonometric functions undo the action of sin, cos, and tan. Where sin(30°) = 0.5, the inverse is arcsin(0.5) = 30°. They take a ratio as input and return the angle that produces that ratio. They're written as sin⁻¹, cos⁻¹, tan⁻¹ — the "-1" is notation, not an exponent.

What's the difference between arcsin, arccos, and arctan?

Each undoes its corresponding primary function. Use arcsin (sin⁻¹) when you know opposite/hypotenuse and want the angle. Use arccos (cos⁻¹) when you know adjacent/hypotenuse. Use arctan (tan⁻¹) when you know opposite/adjacent. The choice depends on which two sides you have.

Why are arcsin and arccos limited to inputs between -1 and 1?

Because sine and cosine themselves only output values in [-1, 1]. There's no real angle whose sine is 2 — that's why our calculator returns an error when you enter a value outside the valid range. Arctan, on the other hand, accepts any real number.

What ranges do the inverse functions return?

By convention, arcsin returns angles in [-90°, 90°], arccos returns angles in [0°, 180°], and arctan returns angles in (-90°, 90°). These "principal value" ranges ensure each input maps to exactly one angle.

How do I find an angle for a value outside the principal range?

The principal value is one solution; others come from the periodicity of trig functions. For sine: if arcsin(v) = θ, then 180° - θ also works (and add 360° multiples). For cosine: if arccos(v) = θ, then -θ + 360°k. For tangent: arctan(v) + 180°k.

Are sin⁻¹ and 1/sin the same thing?

No — and this is one of the most common mistakes in trig. sin⁻¹(x) means "the angle whose sine is x" (the inverse function). 1/sin(x) is the cosecant function, csc(x). They give totally different results. Despite the notation, the "-1" exponent on a function name means inverse, not reciprocal.

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